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C o m b i n i n g T r uc u c k Po w e r A loaded truck (truck 1) has a maximum engine power attain a maximum speed
and is able to
. Another truck (truck 2) has a maximum engine power
and
can attain a maximum speed of . The two trucks are then connected by a long cable, as shown. To solve this problem, assume that each truck, when not attached to another truck, has a speed that is limited only by wind resistance. Also assume (not very realistically) A) That the wind resistance is a constant force (a different constant for each truck though). i.e. It is independent of the speed at which the truck is going. B) That the wind resistance force on each truck is the same before and after the cable is connected, and, C) That the power that each truck's engine can generate is independent of the truck's speed.
P ar ar t A
Find , the maximum speed of the two trucks when they are connected, assuming both engines are running at maximum power. Hint A.1
Method for solving
Hint not displayed
Hint A.2
Re s i s t a n c e f o r c e o n t r u c k 1
Hint not displayed
Hint A.3
N e t w i n d r e s is is t a n c e o n t h e t w o t r u c k s
Hint not displayed
Hint A.4
N et e t p o w e r of of t h e t w o t r u c k s
Hint not displayed
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Hint A.5
Solving for
Hint not displayed E x p r e s s t h e m a x i m u m s p ee ee d i n t e r m s o f
.
ANSWER:
= Correct
Note that truck 1 is going faster when in tow than when under its own power, and that truck 2 is going slower . This is consistent with having the cable connecting the trucks being subject to a tension. Anyone who has ever driven a truck, or closely watched one being driven, will know that this sort of arrangement is very unsafe and consequently is never used. However, train locomotives, which can be coupled together without cables, can combine their power in this way.
D e l i v e r i n g Re R e s c u e Su Su p p l i e s You are a member of an alpine rescue team and must project a box of supplies, with mass , up up an an incline of constant constant slope angle so that that it reaches reaches a stranded stranded skier who who is a vertical distance above the bottom of the incline. The incline is slippery, but there is some friction present, with kinetic friction coefficient . P ar ar t A
Use the work-energy theorem to calculat calculate e the minimum speed box at the bottom of the incline so that it will reach the skier. Hint A.1
that you must give the
H o w t o a p p r o a c h t h e p r o b l em em
In order to use the work-energy theorem, , you need to find an expression for the total work done on the box and for the box's initial and final kinetic energies. At least one of these quantities will depend on the unknown initial speed of the box.
Hint A.2
Fi n d t h e t o t a l w o r k d o n e o n t h e b o x
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What is , the total work done on the box between the moment it is projected and the moment it reaches the skier? Hint A.2.1
Fi n d t h e w o r k d o n e b y g r a v i t y
Hint not displayed
Hint A.2.2
Fi n d t h e w o r k d o n e b y f r i c t i o n
Hint not displayed Express your answ er in term s of som e or all of th e variables and
,
,
,
.
ANSWER:
Hint A.3
Write velocity
= Answer not displayed
Wh a t i s t h e i n i t i a l k i n e t i c e n e r g y ?
, the initial kinetic energy of the box, in terms of the magnitude of its initial and other given variables.
Express your answ er in term s of som e or all of th e variables , and
,
Hint A.4
,
,
= Answer not displayed
Wh a t i s t h e f i n a l k i n e t i c e n e r g y ?
If the box just reaches the skier, what is arrives?
, the kinetic energy of the box when it
Express your answ er in term s of som e or all of th e variables , and
,
.
ANSWER:
,
,
,
,
.
ANSWER:
= Answer not displayed
E x p r e s s y o u r a n s w e r i n t e r m s o f s o m e o r a l l o f t h e v a r i a b le le s and
,
,
,
.
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,
,
MasteringPhysics: Assignment Print View
ANSWER:
= Correct
Dragging a Board A uniform board of length and mass lies near a boundary that separates two regions. In region region 1, the coefficient of kinetic kinetic friction friction between between the board and the surface is , and and in region 2, the coefficient is . The positive direction is shown in the figure.
P ar ar t A
Find the net work done by friction in pulling the board directly from region 1 to region 2. Assume that the board moves at constant velocity. Hint A.1
T h e n et e t f o r c e o f fr fr i c t i o n
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Suppose that the right edge of the board is a distance When the board is at this position, what is the magnitude of the force of friction, ,
from the boundary, as shown.
acting on the board (assuming that it's moving)?
Hint A.1.1
Fract ion of board in region 2
Hint not displayed
Hint A.1.2
Fo r c e o f f r i c t i o n i n r e g i o n 1
Hint not displayed Express th e force ac ting on the board in term s of
ANSWER:
Hint A.2
,
,
,
,
, and
= Answer not displayed
Wo r k a s i n t e g r a l o f f o r c e
After you find the net force of friction
that acts on the board, as a function of
find the net work done by this force, you will need to perform the appropriate work integral,
The lower limit of this integral will be at ANSWER:
.
Upper limit at
. What will be the upper limit?
= Answer not displayed
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, to
MasteringPhysics: Assignment Print View
Hint A.3
D i r e c t i o n o f f o rc rc e o f f r i c t i o n
Don't forget that the force of friction is directed opposite to the direction of the board's motion.
Hint A.4
Fo r m u l a f o r
Ex p r e s s t h e n e t w o r k i n t e r m s o f
,
,
,
, and
.
ANSWER:
= Correct
This answer makes sense because it is as if the board spent half its i ts time in region 1, and half in region 2, which on average, it in fact did.
Pa r t B
What is the total work done by the external force in pulling the board from region 1 to region 2? (Again, assume that the board moves at constant velocity.) Hint B.1
No acc eleration
Hint not displayed Express your answ er in term s of
,
,
,
, and
.
ANSWER:
= Correct
P o w e r D i s s i p a t i o n P u t s a D r a g o n Ra Ra c i n g
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The dominant form of drag experienced by vehicles (bikes, cars, planes, etc.) at operating speeds is called form drag . It increases quadratically with velocity (essentially because the amount of air you run into increases increases with and so does the the amount amount of force you must exert exert on each small volume of air). Thus , where
is the cross-sectional area of the vehicle and
is called the coefficient of drag.
P ar ar t A
Consider a vehicle moving with constant constant velocity drag. Hint A.1
. Find the power power dissipated dissipated by form form
H o w t o a p p r o a c h t h e p r o b l em em
Because the velocity of the car is constant, the drag force is also constant. Therefore, Therefore, you can use the result that the power moving with constant velocity
provided by a constant force
is
to an object
. Be careful to consider the relative
direction of the drag force and the velocity. Ex p r e s s y o u r a n s w e r i n t e r m s o f
,
, and speed
.
ANSWER:
= Correct
P ar ar t B
A certain car has an engine that provides a maximum power . Suppose that the maximum speed of the car, , is limited by a drag force proportional to the square of the speed (as in the previous part). The car engine is now modified, so that the new power is 10 percent greater than the original power (
.
Assume the following: The top speed is limited by air drag. The magnitude of the force of air drag at these speeds is proportional to the square of the speed. By what percentage, , is the top speed of the car increased? ● ●
Hint B.1
F i n d t h e r e l a t i o n s h i p b e t w e e n s p e ed ed a n d p o w e r
Hint not displayed
Hint B.2
H o w i s t h e a l g e b r a d on on e ?
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Hint not displayed E x p r e s s t h e p e r c e n t i n c r e a s e i n t o p s p e e d n u m e r i c a l l y t o t w o s i g n i fi fi c a n t figures.
ANSWER:
= 3 .2 % Correct
You'll note that your answer is very close to one-third of the percentage by which the power was increased. This dependence of small changes on each other, when the quantities are related by proportionalities of exponents, is common in physics and often makes a useful shortcut for estimations.
Work Done by a Spring Consider a spring, with spring constant , one end of which is attached to a wall. The spring is initially unstretched, with the unconstrained end of the spring at position .
Pa r t A
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The spring is now compressed so that the unconstrai unconstrained ned end moves from Using the work integral
to
, find the work done by the spring as it is i s compressed. Hint A.1
Sp r i n g f o r c e a s a f u n c t i o n o f p o s i t i o n
Hint not displayed
Hint A.2
Integrand of the w ork integral
Hint not displayed
Hint A.3
U p pe pe r l i m i t o f t h e w o r k i n t e g r a l
Hint not displayed Ex p r e s s t h e w o r k d o n e b y t h e s p r i n g i n t e r m s o f
and
.
ANSWER:
= Correct
A Ca C a r w i t h C on on s t a n t Po w e r The engine in an imaginary sports car can provide constant power to the wheels over a range of speeds from 0 to 70 miles per hour (mph). At full power, the car can accelerate from zero to 31.0 in time 1.20 . Pa r t A
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.
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At full power, how long would it take for the car to accelerat accelerate e from 0 to 62.0 friction and air resistance. Hint A.1
? Neglect
En e r g y a n d p o w e r
In the absence of friction, the constant power of the engine implies that the kinetic energy of the car increases linearly with time.
Hint A.2
Fi n d t h e r a t i o o f k i n e t i c e n e r g i e s
Find the (numerical) ratio of the car's kinetic energy kinetic energy at time 31.0
at time 62.0
to
, the
.
ANSWER:
= Answer not displayed
Ex p r e s s y o u r a n s w e r i n s e c o n d s .
ANSWER:
4.80 Correct
Of course, neglecting friction, especially air friction, is completely unrealistic at such speeds.
P ar ar t B
A more realistic car would cause the wheels to spin in a manner that would result in the ground pushing it forward with a constant force (in contrast to the constant power in Part A). If such a sports car went from zero to 31.0 in time time 1.20 1.20 , how how long would it take take to go from zero to 62.0 Hint B.1
?
H o w t o a p p r o a c h t h e p r o b l em em
Hint not displayed Ex p r e s s y o u r a n s w e r n u m e r i c a l l y , i n s e c o n d s .
ANSWER:
2.40 Correct
This is probably the first and last time you will come across an imaginary car that goes slower than the real one!
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T h e W or o r k D o n e i n P u l l i n g a Su Su p e r t a n k e r Two tugboats pull a disabled supertanker. Each tug exerts a constant force of 2.20×10 6
,
one at at an angle angle 16.0 16.0 west of of north, north, and and the other other at at an angle angle 16.0 16.0 east of of north, north, as they they pull the tanker a distance 0.680
toward the north.
P ar ar t A
What is the total work done by the two tugboats on the supertanker? Hint A.1
H o w t o a p p r o a c h t h e p r o b l em em
There are two ways to calculate the total work done on an object when several forces act on it. You can compute the quantities of work done on the object by each force and then add them together. Alternatively, you can compute the work done on the object by the net force acting on it. The hints that follow are meant to help you to calculate the total work using the first method.
Hint A.2
Fi n d t h e w o r k d o n e b y o n e t u g b o a t
Hint not displayed E x p r e s s y o u r a n s w e r i n j o u l e s , t o t h r e e s i g n i f i c a n t f i g u r e s. s.
ANSWER:
2.88×10 9 All attempts used; correct answer displayed
P SS SS 7 . 2 Pr Pr o b l e m s U s i n g M e c h a n i c a l En En e r g y I I L e a r n i n g G o a l : To practice Problem-Solving Strategy 7.2 Problems Using Mechanical Energy II.
The Grea Greatt Sandini Sandini is a 60.0 60.0-circus circu s perform performer er who who is shot from a cannon cannon (actu (actually ally a spring gun). You don ’t find many men of his caliber, so you help him design a new gun. This new gun has a very large spring with a very small mass and a force constant of 1100 that he will compress compress with a force of 4400 4400
. The inside of the gun gun barrel is coated with
Teflon, so the average friction force will be only 40.0 during the 4.00 he moves in the barrel. At what speed will he he emerge from the end of the barrel, 2.50 above his initial rest position? P r o b l e m -S -S ol ol v i n g S t r a t e g y : Pr o b l e m s u s i n g m e c h a n i c a l e n e r g y I I IDENTIFY the relevant concepts: The energy approach is useful in solving problems that involve elastic forces as well as gravitational forces, provided the additional additional elastic potential energy is included i ncluded in the http://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=1465961(11 http://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=1465961 (11 of 74) [12/13/2010 7:03:11 PM]
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potential energy . SET UP the problem using the following steps: 1. Decide what the the initial and final states of the the system are. Use the subscript subscript 1 for the initial state and the subscript 2 for the final state. It helps to draw sketches. 2. Define your coordinate system, particularly the level at which . We suggest that you always choose the positive y direction to be upward because this is what assumes. 3. Identify all forces forces that do work, work, including those those that can can ’t be described in terms of potential energy. A free-body diagram is always helpful. 4. List the unknown unknown and known quantities, quantities, including the coordinates coordinates and velocities velocities at each point. Decide which unknowns are your target variables. EXECUTE the solution as follows: Write expressions for the initial and final kinetic and potential energies—that is, , , , and
. The potential energy
now includes both the gravitati gravitational onal potential energy
and the elastic potential energy
, where
is the displacement of the
spring from its unstretched length . Then, relate the kinetic and potential energies and the work done by other forces, , using . If no other forces do work, this expression becomes
. It’s helpful to draw bar graphs
showing the initial and final values of , , and . Then, solve to find whatever unknown quantity is required. EVALUATE your answer: Check whether your answer makes physical sense. Keep in mind that the work done by the gravitational and elastic forces is accounted for by their potential energies; the work of the other forces, , has to be included separately. I D EN EN T I F Y t h e r e l e v a n t c o n c e p t s
The problem involves a spring gun. Therefore, to account for the potential energy associated with its elastic force, the energy approach might be the best method to solve this problem. S ET ET U P t h e p r o b l e m u s i n g t h e f o l l o w i n g s t e p s
P ar ar t A
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Below is a sketch of the initial state of the situation described in this problem. Draw the most suitable set of coordinate axes for this problem. Note that even though you can choose the level to be wherever you like, in most situations it is best to set the zero height to coincide with either the initial or final position, so that the calculations for the gravitational potential energy become easier. For this reason, in i n this particular problem place the origin of your coordinate axes on the black dot marking the performer's initial position. Draw only the positive portion of the coordinate axes. D r a w t h e v e c t o r s s t a r t i n g a t t h e b l a c k d o t . Th T h e l o c a t i o n a n d o r ie ie n t a t i o n o f t h e v e c t o r s w i l l b e g r a d e d . T h e l e ng n g t h o f t h e v e c t o r s w i l l n ot ot b e g r a d e d .
ANSWER:
View Correct This is the coordinate system used in the rest of this problem. Note that since the origin of the axes coincides with the location of the performer's feet, all vertical distances are calculated relative to his feet, and not relative to his center of mass. Now, draw a sketch for the final state showing the performer at the exit of the gun barrel, and identify all the forces that do work on the performer as he travels from the initial to the final state.
Pa r t B
Below is a list of variables representing some of the relevant quantities in this problem. Which ones are known quantities? Ch e c k a l l t h a t a p p l y .
ANSWER:
, magnitude of compressing force
, final heig ht
, m a g n i t u d e o f f r ic ic t i o n
, fo fo r c e c o n s t a n t o f s p r i n g
, mass of body in motion
, di d i s t a n c e t r a v e le l e d b e t w e e n i n it it i a l a n d f i n a l s t a t e
, initial height
, initial speed , final speed
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All attempts used; correct answer displayed Now, make sure that you list all the known quantities on your sketches for the initial and final states of the system. You have identifi identified ed only one unknown, , the final speed of the performer. This is your target variable. However, as you work through the next part, you will find that there may be other unknown quantities that need to be found in order to solve the problem.
E XE XEC U T E t h e s o l u t i o n a s f o l l o w s
P ar ar t C
At what speed Hint C. C.1 1
will The Great Sandini emerge from the end of the gun barrel?
F i n d e x p r e s s i o n s f o r t h e p e r f o r m e r ’s i n i t i a l a n d f i n a l k i n e t i c energies
Hint not displayed
Hint C. C.2 2
F i n d t h e p e r f o r m e r ’s i n i t i a l a n d f i n a l g r a v i t a t i o n a l p o t e n t i a l energies
Hint not displayed
Hint C. C.3 3
F i n d t h e i n i t i a l a n d f i n a l e la la s t i c p o t e n t i a l e n e r g i e s
Hint not displayed
Hint C. C.4 4
Find
Hint not displayed E x p r e s s y o u r a n s w e r i n m e t e r s p e r s e c o n d t o f o u r s i g n if i f i c a n t f i g u r es es .
ANSWER:
15.46 = All attempts used; correct answer displayed
E VA VA L U A T E y o u r a n s w e r
P ar ar t D
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To evaluate whether your result makes sense, it's useful to use bar graphs showing the initial and final values of kinetic and potential energies. These graphs will help you verify whether energy is conserved. The picture to the right is a bar graph showing the initial values of potential energy (gravitational potential energy + elastic potential energy), kinetic energy , and total energy . Which of the following graphs shows the correct final values for , , and ?
ANSWER:
Diagram B Correct
According to your calculations, the total energy decreases by 160 . You can verify that this equals the amount of energy lost to friction, which you previously computed as . So your results make sense. The initial elastic potential energy is for the most part transformed into gravitational and kinetic energy, with a small loss due to friction. In the absence of friction, energy would be conserved and The Great Sandini would emerge from the end of the barrel at an even higher speed.
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A M a s s -S -S p r i n g Sy S y s t e m w i t h Re R e c o i l a n d Fr Fr i c t i o n An object of mass is traveling on a horizontal surface. There is a coefficient of kinetic friction between the object and the surface. The object has speed when it reaches and encounters a spring. The object compresses the spring, stops, and then recoils and travels in the opposite direction. When the object reaches trip, it stops.
on its return
P ar ar t A
Find , the spring constant. Hint A.1
Wh y d o e s t h e o b j e c t s t o p ?
Why does the object come to rest when it returns to
?
Although more than one answ er may be true of the system , you must c h o o s e t h e a n s w e r t h a t e x p l a i n s why t h e o b j ec ec t u l t i m a t e l y c o m e s t o a stop.
ANSWER:
Wh e n t h e o b j e c t r e a c h e s the second tim e all of i t s i n i t i a l e n e rg rg y h a s g o n e i n t o t h e c o m p r e s s i o n a n d e x t e n s i o n of of t h e s p r i n g . Wh e n t h e o b j e c t r e a c h e s the second tim e all of i t s i n i t i a l e n e rg r g y h a s b e e n d i s si si p a t e d b y f r i c t i o n . i s a n e q u il i l i b r i um u m p o s i t i o n a n d a t t h i s p o in in t t h e s p r i n g ex e x e r t s n o f o r c e o n t h e o b j ec ec t . At t h e f o r c e o f f r i c t i o n e x a c t l y b a la la n c e s t h e f o r c e e x e r t e d b y t h e s p r i n g on on t h e o b j e c t . Correct
Hint A.2
H o w d o es e s f r i c t i o n a f fe fe c t t h e s y s t e m ?
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Indicate whether the following statements regarding friction are true or false. Ch e c k a l l t h a t a p p l y .
ANSWER:
Wo r k d o n e b y f r i c t i o n i s e q u a l t o , w here is the mass of an object, is the magnitude of the acc eleration due to gravity, is the coefficient of kinet ic frict ion, and is the distance the object has traveled. E n e r g y d i s s ip ip a t e d b y f r i c t i o n i s e q u a l t o , w here is the coefficient of frict ion, is the acc eleration due to gravity, is the mass of the o b j e c t , a n d i s t h e a m o u n t o f t i m e (s i n c e e n c o u n t e r i n g t h e s p r i n g ) t h e o b j e c t h a s b e e n m o v i n g. g. Fr i c t i o n i s a c o n s e r v a t i v e f o r c e . W o r k d o n e b y f r i c t i o n i s e x a c t l y e q u a l t o t h e n e g at at i v e o f t h e e n e r g y d i s s i p at at e d b y f r i c t i o n .
Correct
Hint A.3
En e r g y s t o r e d i n a s p r i n g
The potential energy stored in a spring having constant
that is compressed compressed a distance distance
is .
Hint A.4
C o m p u t e t h e c o m p r e s s i on on o f t h e s p r i n g
By what distance Hint A.4.1
does the object compress the spring?
H o w t o a p pr pr o a c h t h i s q u e s t i o n
Use the fact that to solve for the distance the spring was compressed. Look at the initial condition when the object originally hits the spring and the final condition when the object returns to .
Hint A.4.2
The value of
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In its final position, the object is not moving. Also the spring is not compressed. Therefore .
Hint A.4.3
Find
What is the value of Hint A.4. A.4.3.1 3.1
?
How to approach this part
Hint not displayed Express your answ er in term s of som e or all of th e variables and
,
,
, and
,
,
, and
, the acc eleration due to gravity.
ANSWER:
= Correct
Hint A.4.4
Find
What is the value of Hint A.4. A.4.4.1 4.1
?
How to approach this part
Hint not displayed Express your answ er in term s of som e or all of th e variables and
, the acc eleration due to gravity.
ANSWER:
= Correct
Express
in te rms of
,
, and
.
ANSWER:
= Correct
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Hint A.5
Pu t t i n g i t a l l t o g e t h e r
In the previous part, at the two ends of the motion considered, the spring had no energy, so was not part of the equation. However, you were able to find a relation for in terms of the known known quantities. quantities. To obtain an equation equation invo involving lving , use conservation of energy again, , but this time, take the initial condition to be the moment when the spring is at its maximum compression and the final condition to be the moment when the spring returns to . So now can be written in terms of and other variables variables..
Hint A.6
The value of
The value of
Hint A.7
is again zero.
Fi n d
for this part of the mot ion
What is the value of Hint A.7.1
for this part of the motion?
How to approach this part
Hint not displayed Ex p r e s s y o u r a n s w e r i n t e r m s o f
and
end up w ith an equation cont aining
, t h e s p r i n g c o n s t a n t , s o t h a t yo yo u
.
ANSWER:
= Correct
Hint A.8
Fi n d
for this part of the mot ion
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What is the value of Hint A.8.1
for this part of the motion?
How to approach this part
Hint not displayed Ex p r e s s y o u r a n s w e r i n t e r m s o f gravity.
,
,
, and
, the acc eleration due to
ANSWER:
=
Express
in term s of
,
,
Correct
, and
.
ANSWER:
= Correct
B u n g ee ee J u m p i n g Kate, a bungee jumper, wants to jump off the edge of a bridge that spans a river below. Kate has a mass , and the surface of the bridge is a height above the water. The bungee cord, which has length when unstretched unstretched,, will will first straighten and then stretch as Kate falls. Assume the following: The bungee cord behaves as an ideal spring once it begins to stretch, with spring constant . Kate doesn't actually jump but simply steps off the edge of the bridge and falls straight downward. Kate's height is negligible compared to the length of the bungee cord. Hence, she can be treated as a point particle. ●
●
●
Use
for the magnitude of the accelerat acceleration ion due to gravity.
Pa r t A
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How far below the bridge will Kate eventually be hanging, once she stops oscillating and comes finally to rest? Assume that she doesn't touch the water. Hint A.1
D e c i d e h o w t o a p p r o a c h t h e p r o b le le m
Here are three possible methods for solving this problem: a. No nonconservative nonconservative forces are acting, so mechanical mechanical energy energy is conserved. Set Kate's gravitational potential potential energy at the top of the bridge equal to the spring potential energy in the bungee cord (which depends on the cord's final length ) and solve for . b. Since nonconservative nonconservative forces are acting, mechanical mechanical energy energy is not conserved. Set the spring potential energy in the bungee bungee cord cord (which depends on ) equal to Kate's Kate's gravitational potential energy plus the work done by dissipative forces. Eliminate the unknown work, and solve for . c. When Kate comes comes to rest she has zero acceleration, acceleration, so the net force force acting on her must be zero. Set the spring force due to the bungee cord (which depends on ) equal to the force of gravity and solve for
.
Which of these these options is the simplest, most accurate way to find find available? ANSWER:
given the information
a b c Correct
Hint A.2
Co m p u t e t h e f o r c e d u e t o t h e b u n g e e c o r d
When Kate is at rest, what is the magnitude exerts on her? Hint A.2.1
of the upward force the bungee cord
Fi n d t h e e x t e n s i o n o f t h e b u n g e e c o r d
Hint not displayed
Hint A.2.2
Fo r m u l a f o r t h e f o r c e d u e t o a s t r e t c h e d c o r d
Hint not displayed Ex p r e s s y o u r a n s w e r i n t e r m s o f t h e c o r d ' s f i n a l s t r e t c h e d l e n g t h and q u a n t i t i e s g i v e n i n t h e p r o b l e m i n t r o d u c t i o n . Yo Y o u r a n s w e r s h o u l d no no t depend on Kat e's mass .
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ANSWER:
= Correct
Set this force equal to Kate's weight, and solve for
.
E x p r e s s t h e d i s t a n c e i n t e r m s o f q u a n t i t i e s g i v e n in in t h e p r o b l e m introduction.
ANSWER:
= Correct
P ar ar t B
If Kate just touches the surface of the river on her first downward trip (i.e., before the first bounce), what is the spring constant constant ? Ignore Ignore all all dissipative forces. Hint B.1
D e c i d e h o w t o a p p r o a c h t h e p r o b le le m
Here are three possible methods for solving this problem: a. Since nonconservative nonconservative forces are ignored, ignored, mechanical energy energy is conserved. Set Kate's gravitational potential energy at the top of the bridge equal to the spring potential energy in the bungee cord at the lowest point (which depends on ) and solve for . b. Nonconservat Nonconservative ive forces can be ignored, ignored, so mechanical energy energy is conserved. Set the spring potential potential energy energy in the bungee bungee cord cord (which depends on ) equal to Kate's Kate's gravitational potential potential energy at the top of the bridge plus the work done by gravity as Kate falls. falls. Compute Compute the work work done done by by gravity, then solve for . c. When Kate is being held held just above the water water she has zero acceleration, acceleration, so the net force acting on her must be zero. Set the spring force due to the bungee cord (which depends on ) equal to the force of gravity and solve for .
Which of these options is the simplest, most accurate way to find find available? ANSWER:
given the information
a b c Correct
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Hint B.2
Fi n d t h e i n i t i a l g r a v i t a t i o n a l p o t e n t i a l e n e r g y
What is Kate's gravitational potential energy at the moment she steps off the bridge? (Define the zero of gravitational potential to be at the surface of the water.) E x p r e s s y o u r a n s w e r i n t e r m s o f q u a n t i t i e s g i v e n in in t h e p r o b l e m introduction.
ANSWER:
=
Hint B.3
Correct
Fi n d t h e e l a s t i c p o t e n t i a l e n e r g y i n t h e b u n g e e c o r d
What is the elastic potential energy stored in the bungee cord when Kate is at the lowest point of her first downward trip? Hint B.3.1
Fo r m u l a f o r e l a s t i c p o t e n t i a l e n e r g y
The elastic potential energy of the bungee cord (which we are treating as an ideal spring) is , where
is the amount by which the cord is stretched beyond its unstretched length.
Hint B.3.2
H o w m u c h i s t h e b un u n g ee e e c o r d s t r e t c h e d? d?
By how much is the bungee cord stretched when Kate is at a depth bridge? Ex p r e s s y o u r a n s w e r i n t e r m s o f
and
below the
.
ANSWER:
=
Correct
E x p r e s s y o u r a n s w e r i n t e r m s o f q u a n t i t i e s g i v e n in in t h e p r o b l e m introduction.
ANSWER:
= Correct
Express
in term s of
,
,
, and
.
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ANSWER:
= Correct
Ci r c l i n g B a l l A ball of mass is attached to a string of length . It is being swung in a vertical circle with enough speed so that the string remains taut throughout the ball's motion. Assume that the ball travels freely in this vertical circle with negligible loss of total mechanical energy. At the top and bottom of the vertical circle, the ball's speeds are and , and the corresponding correspondin g tensions in the string are
and
magnitudes
.
and and
have .
P ar ar t A
Find , the difference between the magnitude of the tension in the string at the bottom relative to that at the top of the circle. Hint A.1
How to approach this problem
Hint not displayed
Hint A.2
F i n d t h e s u m o f f or or c e s a t t h e b o t t o m o f t h e c i r c l e
Hint not displayed
Hint A.3
F i n d t h e a c c e l e r a t i on on a t t h e b o t t o m o f t h e c i r c l e
Hint not displayed
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Hint A.4
F i n d t h e t e n s i o n at a t t h e b ot ot t o m o f t h e c i r c l e
Hint not displayed
Hint A.5
Fi n d t h e s u m o f f o r c e s a t t h e t o p o f t h e c i r c l e
Hint not displayed
Hint A.6
F i n d t h e a c c e l e r a t i o n at at t h e t o p o f t h e c i r c l e
Hint not displayed
Hint A.7
Fi n d t h e t e n s i o n a t t h e t o p o f t h e c i r c l e
Hint not displayed
Hint A.8
Fi n d t h e r e l a t i o n s h i p b e t w e e n
and
Hint not displayed E x p r e s s t h e d i f f e re re n c e i n t e n s i o n i n t e r m s o f s h o u l d not a p p e a r i n y o u r f i n a l a n s w e r .
and
. The quantities
and
ANSWER:
=
Correct
The method outlined in the hints is really the only practical way to do this problem. If done properly, finding the differenc difference e between the tensions, , can be accomplished fairly simply and elegantly.
Dr a g o n a Sk y d i v e r
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A skydiver of mass jumps from a hot air balloon and falls a distance before reaching a terminal velocity of magnitude . Assume that the magnitude of the accelerat acceleration ion due to gravity is . Pa r t A
What is the work air? Hint A.1
done on the skydiver, over the distance
, by the drag force of the
H o w t o a p p r o a c h t h e p r o b l em em
Hint not displayed
Hint A.2
Fi n d t h e c h a n g e i n p o t e n t i a l e n e r g y
Hint not displayed
Hint A.3
Fi n d t h e c h a n g e i n k i n e t i c e n e r g y
Hint not displayed Ex p r e s s t h e w o r k i n t e r m s o f acc eleration due to gravity
,
,
, and the magnitude of the
.
ANSWER:
= Correct
P ar ar t B
Find the power velocity . Hint B.1
supplied by the drag force after the skydiver has reached terminal
H o w t o a p p r o a c h t h e p r o b l em em
Hint not displayed
Hint B.2
Magnitude of the drag force
Hint not displayed
Hint B.3
Re l a t i v e d i r e c t i o n o f t h e d r a g f o r c e a n d v e l o c i t y
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Ex p r e s s t h e p o w e r i n t e r m s o f q u a n t i t i e s g i v e n i n t h e p r o b l e m i n t r o d u c t i o n .
ANSWER:
=
Correct
E n e r g y i n a Sp Sp r i n g G r a p h i n g Q u e s t i o n A toy car is held at rest against a compressed spring, as shown in the figure. When released, the car slides across the room. Let be the initial position of the car. Assume that friction is negligible.
P ar ar t A
Sketch a graph of the total energy of the spring and car system. There is no scale given, so your graph should simply reflect the qualitative shape of the energy vs. time plot. ANSWER:
View All attempts used; correct answer displayed
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P ar ar t B
Sketch a plot of the elastic potential energy of the spring from the point at which the car is released to the equilibrium position of the spring. Make your graph consistent with the given plot of total energy (the gray line l ine given in the graphing window). Hint B.1
De t e r m i n e t h e s i g n o f t h e i n i t i a l e l a s t i c p o t e n t i a l e n e r g y
At the instant the car is released, the spring is compressed. Therefore, is the spring's initial elastic potential energy positive, negative, or zero? ANSWER:
positive negative zero Correct
Hint B.2
De t e r m i n e t h e s i g n o f t h e i n i t i a l k i n e t i c e n e r g y
Is the initial kinetic energy of the cart positive, negative, or zero? ANSWER:
positive negative zero Correct
Hint B.3
De t e r m i n e t h e s i g n o f t h e f i n a l e l a s t i c p o t e n t i a l e n e r g y
When the car reaches the equilibrium position of the spring, is the elastic potential energy positive, negative, or zero? ANSWER:
positive negative zero Correct
Hint B.4
T h e s h a p e of of t h e e l a s t i c p o t e n t i a l e n e r g y g r a p h
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The elastic potential energy of a spring with spring constant compressed to position is given by
that is stretched stretched or
, where
is the equilibrium position of the spring.
ANSWER:
View Correct
P ar ar t C
Sketch a graph of the car's kinetic energy from the moment it is released until it passes the equilibrium position of the spring. Your graph should be consistent with the given plots of total energy (gray line in graphing window) and potential energy (gray parabola in graphing window). ANSWER:
View Correct
F u n w i t h a S pr p r i n g G un un
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A spring-loaded toy gun is used to shoot a ball of mass
straight up in the air,
as shown in the figure. The spring has spring constant
. If the spring is
compressed a distance of 25.0 centimeters from its equilibrium position and then released, the ball reaches a maximum height (measured from the equilibrium position of the spring). There is no air resistance, and the ball never touches the inside of the gun. Assume that all movement occurs in a straight line li ne up and down along the y axis.
P ar ar t A
Which of the following statements are true? Hint A.1
N o n c o n s e r v at at i v e f o r c e s
Hint not displayed
Hint A.2
Fo r c e s a c t i n g o n t h e b a l l
Hint not displayed Ch e c k a l l t h a t a p p l y .
ANSWER:
M e c h a n i c a l e n e r g y i s c o n s e r ve ve d b e c a u s e n o d i s s i pa p a t i v e f o r c e s p e r f o r m w o r k o n t h e b a ll ll . The forces of gravity and the spring have potential e n er e r g i es es a s s o c i a t e d w i t h t h e m . N o c o n s e r v a t i v e f or o r c e s a c t i n t h i s p r ob ob l e m a f t e r t h e ball is released from t he spring gun.
Correct
P ar ar t B
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Find the muzzle velocity of the ball (i.e., the velocity of the ball at the spring's equilibrium position ). Hint B.1
D e t e r m i n e h ow o w t o a p p r o a c h t h e p r o bl bl e m
Hint not displayed
Hint B.2
E n er er g y e q u a t i o n s
Hint not displayed
Hint B.3
D e t e r m i n e w h i c h t w o l oc oc a t i o n s y o u s h o u l d e x a m i n e
Hint not displayed
Hint B.4
Fi n d t h e i n i t i a l e n e r g y o f t h e s y s t e m
Hint not displayed
Hint B.5
Determ ine the final energy
Hint not displayed
Hint B.6
Creating an equation
Hint not displayed ANSWER:
= 4.78 Correct
P ar ar t C
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Find the maximum height Hint C. C.1 1
of the ball.
Ch o o s e t w o l o c a t i o n s t o e x a m i n e
Hint not displayed
Hint C. C.2 2
Fi n d t h e i n i t i a l e n e r g y
Hint not displayed
Hint C. C.3 3
De t e r m i n e t h e f i n a l e n e r g y
Hint not displayed
Hint C. C.4 4
Cr e a t i n g a n e q u a t i o n
Hint not displayed Ex p r e s s y o u r a n s w e r n u m e r i c a l l y , i n m e t e r s .
ANSWER:
= 1.17 Correct
In this problem you practiced applying the law of conservation of mechanical energy to a physical situation to find the muzzle velocity and the maximum height reached by the ball.
P ar ar t D
Which of the following actions, if done independe independently, ntly, would increase the maximum height reached by the ball? Ch e c k a l l t h a t a p p l y .
ANSWER:
reducing the spring constant
increasing the spring constant d e c r e a s i n g t h e d i s t a n c e t h e s p r i ng ng i s c o m p r e s s e d
i n c r e a s i n g t h e d i s t a n c e t h e s p r i ng ng i s c o m p r e s s e d
decreasing the mass of t he ball increasing the mass of t he ball t i l t i n g t h e s p r i ng ng g u n s o t h a t i t i s a t a n a n g l e d e g r e es e s f r o m t h e h o r i zo zo n t a l
Correct
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G r a p h i n g G r a v i t a t i o n a l Po P o t e n t i a l En En e r g y A 1.00
ball is thrown directly upward with an initial speed of 16.0
.
A graph of the ball's gravitational potential energy vs. height, , for an arbitrary initial velocity is given in Part A. The zero point of gravitational potential energy is located at the height at which the ball leaves the thrower's hand. For this problem, take as the acceleration due to gravity. P ar ar t A
Draw a line on the graph representing the total energy Hint A.1
of the ball.
H o w t o a p p r o a c h t h e p r o b l em em
The total energy is the sum of the kinetic energy and potential energy. You can compute the total energy at any point in the ball's trajectory, but the simplest method is to add the initial kinetic and potential energies just as the ball is thrown.
Hint A.2
Fi n d t h e i n i t i a l k i n e t i c e n e r g y
When the ball first leaves the thrower ’s hand, what is its kinetic energy ANSWER:
Hint A.3
= 128 Correct
Fi n d t h e i n i t i a l p o t e n t i a l e n e r g y
What is the potential energy ANSWER:
Hint A.4
?
of the ball when it first leaves the thrower's hand?
= Answer not displayed
Sh a p e o f t h e t o t a l e n e r g y g r a p h
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As the ball ascends, does its total energy increase, decrease, or stay the same? ANSWER:
increase decrease stay the same Correct
The law of conservation of energy guarantees that the total energy of the ball remains constant throughout its motion. The increase in potential energy as the ball ascends is exactly balanced by the decrease in its kinetic energy.
ANSWER:
View Correct
P ar ar t B
Using the graph, determine the maximum height reached by the ball. Hint B.1
Maximum height
The ball reaches its maximum height when its velocity (and therefore kinetic energy) is zero, so all of its energy is potential. This occurs at the height at which the total energy and potential energy graphs intersect. The ball does not have enough energy to rise above this point on the potential energy graph. Ex p r e s s y o u r a n s w e r t o o n e d e c i m a l p l a c e .
ANSWER:
12.8 Correct
The ball reaches its maximum height when its velocity (and therefore kinetic energy) is zero, so all of its energy is potential. This occurs at the height at which the total energy and potential energy graphs intersect.
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P ar ar t C
Draw a new gravitational potential energy vs. height graph to represent the gravitational potentia pote ntiall ener energy gy if the ball had a mass of 2.00 . The grap graph h for a 1.00 1.00-ball with an arbitrary initial velocity is provided again as a reference. Take as the acceleration due to gravity. Hint C. C.1 1
Slope
The gravitational potential energy is defined by In a graph of potential energy vs. height,
Hint C. C.2 2
. is the slope.
De t e r m i n e t h e n e w g r a v i t a t i o n a l p o t e n t i a l e n e r g y
Whatt is the gra Wha gravit vitati ationa onall pot potent ential ial en energ ergy y for a 2.0 2.000Take
ballll at a hei ba heigh ghtt of
?
as the acc eleration due to gravity and express your
a n s w e r t o t h r e e d e c i m a l p l ac ac e s .
ANSWER:
= 100 Correct
The new graph of potential energy versus height must pass through the point .
ANSWER:
View Correct For a ball with twice the mass, you should expect the plot of potential energy vs. height to have twice the slope.
K i n e t i c a n d Po P o t e n t i a l E n e r g y o f B a s e b a l l Gr G r a p h i n g Qu Qu e s t i o n
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A baseball is thrown directly upward at time and is caught again at time . Assume that air resistance is so small that it can be ignored and that the zero point of gravitational potential potential energy is located at the position at which the ball leaves the thrower's hand. P ar ar t A
Sketch a graph of the kinetic energy of the baseball. Hint A.1
De t e r m i n e t h e s i g n o f t h e i n i t i a l k i n e t i c e n e r g y
Hint not displayed
Hint A.2
T h e s h a p e of o f t h e k i n e t i c e n e r g y g r ap ap h
Hint not displayed ANSWER:
View All attempts used; correct answer displayed
P ar ar t B
Based on the graph of kinetic energy given (gray curve in the graphing window), sketch a graph of the baseball's gravitational potential energy. Hint B.1
I n i t i a l g r a v i t a t i o n a l p o t e n t i a l e n e r gy gy
Hint not displayed
Hint B.2
The shape of the gravitat ional potential energy graph
Hint not displayed
Hint B.3
Using conservat ion of energy
Hint not displayed
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ANSWER:
View Correct
P ar ar t C
Based on the kinetic and potential energy graphs given, sketch a graph of the baseball's total energy. Hint C. C.1 1
Total energy
Hint not displayed ANSWER:
View Correct
Loop the Loop
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A roller coaster car may be approximated approximat ed by a block of mass . The car, which starts from rest, is released at a height above the ground and slides along a frictionless track. The car encounters a loop of radius , as shown. Assume that the initial height is great enough so that the car never loses contact with the track.
P ar ar t A
Find an expression for the kinetic energy of the car at the top of the loop. Hint A.1
Fi n d t h e p o t e n t i a l e n e r g y a t t h e t o p o f t h e l o o p
What is the potential energy of the car when it is at the top of the loop? Define the gravitational potential energy to be zero at . Ex p r e s s y o u r a n s w e r i n t e r m s o f
ANSWER:
and other given quantities.
= Answer not displayed
Ex p r e s s t h e k i n e t i c e n e r g y i n t e r m s o f
,
,
, and
.
ANSWER:
=
Correct
Pa r t B
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Find the minimum initial height height at which which the car can can be be released released that still allows allows the car to stay in contact with the track at the top of the loop. Hint B.1
How to approach this part
Meaning of "stay in contact" For the car to just stay in contact through the loop, without falling, the normal force that acts on the car when it's at the top of the loop must be zero (i.e., ). Find the velocity at the top such that the remaining force on the car i.e. its weight provides the necessary centripetal acceleration. If the velocity were any greater, you would additionally require some force from the track to provide the necessary centripetal acceleration. If the velocity were any less, the car would fall off the track. Use the above described condition to find the velocity and then the result from the above part to find the required height.
Hint B.2
A c c e l e r a t i o n at a t t h e t o p o f t h e l oo oo p
Hint not displayed
Hint B.3
Normal force at t he top of the loop
Hint not displayed
Hint B.4
Solving for
Hint not displayed Ex p r e s s t h e m i n i m u m h e i g h t i n t e r m s o f
ANSWER:
=
.
Correct
For the car will still complete the loop, though it will require some normal reaction even at the very top. For the car will just oscillate. Do you see this? For , the cart will lose contact with the track at some earlier point. That is why roller coasters must have a lot of safety features. If you like, you can check that the angle at which the cart loses contact with the track is given by .
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N o t Q u i t e a r o u n d t h e Gl Gl o b e A large globe, with a radius of about 5 , was built in Italy between 1982 and 1987. Imagine that such a globe has a radius and a frictionless surface. A small block of mass slides starts from rest at the very top of the globe and slides along the surface of the globe. The block leaves the surface of the globe when it reaches a height above the ground. The geometry geometry of the situation is shown in the figure for an arbitrary height
.
Pa r t A
Consider what happens at the moment when the block leaves the surface of the globe. Which of the following statements are correct? a. The net acceleration acceleration of the the block is directed straight down. down. b. The component of the force of gravity toward the center of the globe is equal to the magnitude of the normal force. c. The force of gravity is the only only force acting on the block.
Hint A.1
H o w i s t h e n o r m a l f or o r c e c h a n g i ng ng ?
Hint not displayed
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ANSWER:
a only b only c o n ly ly a and b a and c b and c a and b and c Correct
P ar ar t B
Which of the following statements is also true at the moment when the block leaves the surface of the globe? ANSWER:
T h e c e n t r i p e t a l a c c e l e r a t i o n i s ze ze r o . T h e n o r m a l f o r c e i s ze ze r o . T h e n et e t a c c e l e r a t i o n o f t h e b l o c k i s p a r a ll ll e l t o i t s velocity. T h e k i n e t i c e n e r g y of o f t h e b l o c k e q u a ls ls i t s p o t e n t i a l energy. Correct
P ar ar t C
Using Newton's 2nd law, find , the speed of the block at the critic critical al moment when the block leaves the surface of the globe. Assume that the height at which the block leaves the surface of the globe is . Hint C. C.1 1
How to approach this problem
Since the normal force goes to zero at the critical moment when the block leaves the surface of the globe, it is the radial component of the gravitational force that generates the entire centripetal acceleration at this point. Use this fact and Newton's 2nd law to relate the acceleration due to gravity and the centripetal acceleration.
Hint C. C.2 2
Fi n d t h e c e n t r i p e t a l a c c e l e r a t i o n
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What is , the magnitude magnitude of of the centripetal centripetal acceleration acceleration of the block when its speed is ? Assume that the block has not lost l ost contact with the globe. Hint C.2.1
Fo r m u l a f o r c e n t r i p e t a l a c c e l e r a t i o n
Hint not displayed Ex p r e s s y o u r a n s w e r i n t e r m s o f
and
.
ANSWER:
= Correct
Hint C. C.3 3
F i n d t h e r a d i a l c o m p o n e n t o f t h e g r a vi v i t a t i o n a l f or or c e
What is , the magnitude of the radial component of the gravitati gravitational onal force on the block when the block is at the position indicated in the figure? Ex p r e s s y o u r a n s w e r i n t e r m s o f
,
, and
.
ANSWER:
=
Hint C. C.4 4
Correct
Wh a t i s
?
Having found the block) and
, you now need to find
in terms of
. You need to find a right triangle where
is the hypoteneu hypoteneuse. se. Using this triangle, what is Gi v e y o u r a n s w e r i n t e r m s o f
and
(the height of
is the included angle and
?
.
ANSWER:
= Correct
Ex p r e s s t h e s p e e d i n t e r m s o f
,
a c c l e r a t i o n d u e t o g r a v i t y . Do D o not u s e
, and
, the magnitude of the
in your answ er.
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ANSWER:
= Correct
P ar ar t D
Use the law of conservation of energy to find . This will give you a difference expression for than you found in the previous part. Hint D. D.1 1
How to apply conservation of energy
The law of conservation of energy states that . You may assume that the initial velocity of the block is negligible, so that the block's initial kinetic energy is zero. The final kinetic energy of the block can be easily expressed in terms of and . The initial and final potential energies of the block simply depend on the height of the block above the ground (or any other reference point). Express
in term s of
,
, and
.
ANSWER:
= Correct
P ar ar t E
Find
, the height from the ground at which the block leaves the surface of the globe.
Hint E. E.1 1
H o w t o a pp p p r o a c h t h i s q u e st st i o n
Hint not displayed Ex p r e s s
in term s of
.
ANSWER:
= Correct
P r o j e c t i l e M o t i o n a n d Co C o n s e r v a t i o n o f En E n e r g y Ra Ra n k i n g T a s k
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P ar ar t A
Six baseball throws are shown below. In each case the baseball is thrown at the same initial speed and from the same height above the ground. Assume that the effects of air resistance are negligible. Rank these throws according to the speed of the baseball the instant before it hits the ground. Hint A.1
H o w t o a p p r o a c h t h e p r o b l em em
Hint not displayed R a n k f r o m l a r g e s t t o s m a l l e s t . T o r a n k i t e m s a s e q u i v a le l e n t , o v e r la la p t h e m .
ANSWER:
View Correct This answer is best understood in terms of conservation of energy. The initial energy of the ball is independen independentt of the direction in which it is thrown. The initial and final potential energies of the ball are the same regardless of the trajectory. Therefore, the final kinetic energy, and therefore the final speed, of the ball must be the same no matter in what direction it is thrown.
Sh o o t i n g a b a l l i n t o a b o x Two children are trying to shoot a marble of mass into a small box using a spring-loaded gun that is fixed on a table and shoots horizontally from the edge of the table. The edge of
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the table is a height above the top of the box (the height of which is negligibly small), and the center of the box is a distance from the edge of the table. The spring has a spring constant . The first child compresses the spring a distance and finds that the marble falls short of its target by a horizontal distance .
P ar ar t A
By what what distance, , should should the second child compress the spring so that the marble lands in the middle of the box? (Assume that height of the box is negligible, so that there is no chance that the marble will hit the side of the box before it lands in the bottom.) Hint A.1
Ge n e r a l m e t h o d f o r f i n d i n g
For this part of the problem, you don't need to consider the first child's toss. (The quantities and should not appear in your answer.) Consider the energy conservation and kinematic kinematic relations relations for the marble, marble, and and solve for its range, of
, ,
, and
Hint A.2
, in terms
.
Initial speed of the m arble
Use conservation of energy to find the initial speed,
, of the second marble.
Ex p r e s s y o u r a n s w e r i n t e r m s o f
.
,
, and
ANSWER:
= Correct
Hint A.3
T i m e f o r t h e m a r b l e t o h i t t h e g r o un un d
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Use kinematics kinematics to find shot off the table.
, the time it takes the second marble to to hit the ground ground after after it is
Ex p r e s s y o u r a n s w e r i n t e r m s o f
and
.
ANSWER:
= Correct
Hint A.4
Co m b i n i n g e q u a t i o n s a n d s o l v i n g f o r
The kinematic equation for the motion along the x axis is expressions for , ,
,
, and
and
. Using the
from the previous hints, solve for
in terms of the quantities
.
Express th e dista nce in te rms of
,
,
,
, and
.
ANSWER:
= Correct
Pa r t B
Now imagine that the second child does not know the mass of the marble, the height of the table above the floor, or the spring constant. Find an expressi expression on for that depends only on and distance measurements. Hint B.1
Compute
Use your answer to Part A to write ANSWER:
Express
in terms of
, , ,
,
, and
.
= Answer not displayed
in te rms of
,
, and
.
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ANSWER:
= Correct
Sh o o t i n g a B l o c k u p a n I n c l i n e A block of mass is placed placed in a smooth-bo smooth-bored red spring gun at the bottom of the incline so that it compresses the spring by an amount . The spring has spring constant . The incline makes an angle with the horizontal and the coefficient of kinetic friction between the block block and the incline is . The block is released, exits the muzzle of the gun, gun, and slides up an incline a total distance .
Pa r t A
Find , the distance traveled traveled along along the incline by the block block after it exits the gun. gun. Ignore friction when the block is inside the gun. Also, assume that the uncompressed spring is while inside of the gun). Use just at the top of the gun (i.e., the block moves a distance for the magnitude of acceleration due to gravity. Hint A.1
H o w t o a p p r o a c h t h e p r o b l em em
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This is an example of a problem that would be very difficult using only Newton's laws and calculus. Instead, use the Work-Energy Theorem: , where is the final energy, is the initial energy, and is the work done on the system by external forces. Let the gravitational potential energy be zero before the spring is released. Then, is the potential energy due to the spring, is the potential energy due to gravity, and this equation completely, solve for
Hint A.2
is the work done by friction. Once you've set up .
Fi n d t h e i n i t i a l e n e r g y o f t h e b l o c k
Find the initial energy of the block. Take the gravitational potential energy to be zero before the spring is released. Hint A.2.1
Po t e n t i a l e n e r g y o f a c o m p r e s s e d s p r i n g
Hint not displayed E x p r e s s y o u r a n s w e r i n t e r m s o f p a r a m e t e r s g i v e n in in t h e p r o b l e m introduction.
ANSWER:
= Correct
Hint A.3
Find Hint A.3.1
Fi n d t h e w o r k d o n e b y f r i c t i o n
, the work done by friction on the block. H ow ow t o c o m p u t e w o r k
Hint not displayed Express
in term s of
,
,
,
, and
.
ANSWER:
= Correct
Hint A.4
Find t he final energy of the bloc k
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Find an expression for the final energy
of the block (the energy when it has
traveled a distance up the incline). Assume that the gravitational potential energy of the block is zero before the spring is released and that the block moves a distance inside of the gun. Hint A.4.1
Wh a t f o r m d o e s t h e e n e r g y t a k e ?
Hint not displayed Your answ er should cont ain
and
.
in term s of
,
ANSWER:
= Correct
Express th e dista nce
,
,
,
, and
.
ANSWER:
= Correct
S l i d i n g I n So So c k s Suppose that the coefficient of friction between your feet and the floor, while wearing socks, is 0.250. Knowing this, you decide to get a running start and then slide across the floor. P ar ar t A
If your speed is 3.00
when you start to slide, what distance
will you slide before
stopping? Ex p r e s s y o u r a n s w e r i n m e t e r s .
ANSWER:
1.84 Correct
P ar ar t B
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Now, suppose that your young cousin sees you sliding and takes off her shoes so that she can slide as well (assume her socks have the same coefficient of friction as yours). Instead of getting a running start, she asks you to give her a push. So, you push her with a force of 125 over a distance of 1.00 . If her mass is 20.0 , what distance does she slide (i.e., how far does she move after the push ends)? Remember that the friction force is acting anytime that she is moving. Hint B.1
H o w t o a p p r o a c h t h e p r o b l em em
Hint not displayed Ex p r e s s y o u r a n s w e r i n m e t e r s .
ANSWER:
= 1.55 Correct
S p r i n g a n d Pr Pr o j e c t i l e A child's toy consists of a block that attaches to a table with a suction cup, a spring connected to that block, a ball, ball, and a launching ramp. The spring has a spring constant , the ball has a mass , and the ramp rises a height above the table, the surface of which is a height above the floor. Initially, the spring rests at its equilibrium length. The spring then is compressed a distance , where the ball is held at rest. The ball is then released, launching it up the ramp. When the ball leaves the launching ramp its velocity vector makes an angle with respect to the horizontal. Throughout this problem, ignore friction and air resistance. P ar ar t A
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Relative to the initial configuration (with the spring relaxed), when the spring has been compressed, the ball-spring system has ANSWER:
g a i n ed ed k i n e t i c e n e r g y gained potent ial energy lost kinetic energy l o s t p o t e n t i a l e n er er g y Correct
P ar ar t B
As the spring expands (after the ball is released) the ball-spring system ANSWER:
g a i n s k i n e t i c e n e r g y a nd nd l o s e s p o t e n t i a l e n e r g y g a i n s k i n e t i c e n e r g y a n d g a in in s p o t e n t i a l e n e r g y l o s e s k i n e t i c e n e r g y a n d ga ga i n s p o t e n t i a l e n e r g y l o s e s k i n e t i c e n e r g y a n d lo lo s e s p o t e n t i a l e n e r g y Correct
P ar ar t C
As the ball goes up the ramp, it ANSWER:
g a i n s k i n e t i c e n e r g y a nd nd l o s e s p o t e n t i a l e n e r g y g a i n s k i n e t i c e n e r g y a n d g a in in s p o t e n t i a l e n e r g y l o s e s k i n e t i c e n e r g y a n d ga ga i n s p o t e n t i a l e n e r g y l o s e s k i n e t i c e n e r g y a n d lo lo s e s p o t e n t i a l e n e r g y Correct
P ar ar t D
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As the ball falls to the floor (after having reached its maximum height), it ANSWER:
g a i n s k i n e t i c e n e r g y a nd nd l o s e s p o t e n t i a l e n e r g y g a i n s k i n e t i c e n e r g y a n d g a in in s p o t e n t i a l e n e r g y l o s e s k i n e t i c e n e r g y a n d ga ga i n s p o t e n t i a l e n e r g y l o s e s k i n e t i c e n e r g y a n d lo lo s e s p o t e n t i a l e n e r g y Correct
P ar ar t E
Which of the graphs shown best represents the potential energy of the ball-spring system as a function of the ball's horizontal displacement? Take the "zero" on the distance axis to represent the point at which the spring is fully compressed compressed.. Keep in mind that the ball is not attached to the spring, and neglect any recoil of the spring after the ball loses contact with it.
ANSWER:
C Correct
P ar ar t F
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Calculate Hint F. F.1 1
, the speed of the ball when it leaves leaves the launching ramp. G e n er er a l a p p r o a c h
Hint not displayed
Hint F. F.2 2
Fi n d t h e i n i t i a l m e c h a n i c a l e n e r g y
Hint not displayed
Hint F. F.3 3
F i n d t h e m e c h a n i c a l e n e r g y at at t h e e n d o f t h e r a m p
Hint not displayed
Hint F. F.4 4
Is energy conserved?
Hint not displayed Express th e speed of th e ball in term s of
,
,
,
,
, a n d /o /o r
.
ANSWER:
= Correct
P ar ar t G
With what speed will the ball hit the floor? Hint G. G.1 1
G e ne ne r a l a p p r o a c h
Hint not displayed
Hint G. G.2 2
I n i t i a l m e c h a n i c a l e n e r gy gy
Hint not displayed
Hint G. G.3 3
Fi n d t h e f i n a l m e c h a n i c a l e n e r g y
Hint not displayed
Hint G. G.4 4
Is energy conserved?
Hint not displayed
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Express th e speed in te rms of
,
,
,
,
, a n d /o /o r
.
ANSWER:
= Correct
Spring Gun A spring-loaded toy gun is used to shoot a ball straight up in the air. The ball reaches a maximum height , measured from the equilibrium position of the spring.
P ar ar t A
The same ball is shot straight up a second time from the same gun, but this time the spring is compressed only half as far before firing. How far up does the ball go this time? Neglect friction. Assume that the spring is ideal and that the distance by which the spring is compressed is negligible compared to . Hint A.1
Po t e n t i a l e n e r g y o f t h e s p r i n g
Hint not displayed
Hint A.2
Po t e n t i a l e n e r g y o f t h e b a l l
Hint not displayed
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ANSWER:
height = Correct
S p r i n g s i n T w o Di Di m e n s i o n s The ends of of two identical springs are connected. connected. Their unstretched lengths
are negligibly negligibly
small and each has spring constant constant . After being being connected, connected, both springs are stretched an an amount and their free ends are anchored at where the springs are connected to each other is now pulled to the position ( , ). Assume that ( , ) lies in the first quadrant.
and
as shown . The point
P ar ar t A
What is the potential energy of the two-spring system after the point of connection has been move moved d to posi position tion ( , )? Keep in mind mind that the unst unstretch retched ed lengt length h of of each each spring spring is much less than Hint A.1
and can be ignored (i.e.,
).
A n i m p o r t a n t p r o p e r t y o f t h e p o t e n t i a l e n e r gy gy
Hint not displayed
Hint A.2
P o t e n t i a l e n e r g y o f t h e l e f t -h -h a n d s p r i n g
Hint not displayed Express th e pot ential in term s of
,
,
, and
.
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ANSWER:
= Correct
P ar ar t B
Find the force
on the junction point, the point where the two springs are attached to
each other. Hint B.1
H o w t o a p p r o a c h t h e p r o b l em em
Hint not displayed
Hint B.2
Co m p o n e n t s o f t h e f o r c e v e c t o r
Hint not displayed Ex p r e s s
as a vect or in term s of the unit vect ors
and
.
ANSWER:
=
Correct
Notice how much more difficult it would have been to obtain the force via vi a vector addition (computing the two components of the force from each spring, then adding them). This is the power of scalar potential functions: They allow you to simply add up the contributions, without having to worry about vectors or coordinate axes. By taking the gradient of the potential, you automatica automatically lly obtain the desired vector quantities.
W o r k a n d Po P o t e n t i a l En E n e r g y o n a Sl S l i d i ng ng B l o c k w i t h Fr i c t i o n
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A block of weight sits on a plane inclined at an angle kinetic friction between the plane and the block is .
A force
as shown. The coefficient of
is applied to push the block up the incline at constant speed.
P ar ar t A
What is the work distance
done on the block by the force of friction as the block moves a
up the incline?
Hint A.1
A formula for work
Hint not displayed
Hint A.2
F i n d t h e m a g n i t u d e o f t h e f r i c t i o n a l f or or c e
Hint not displayed Ex p r e s s y o u r a n s w e r i n t e r m s o f s o m e o r a l l o f t h e f o l l o w i n g :
,
,
,
ANSWER:
=
Correct
P ar ar t B
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.
MasteringPhysics: Assignment Print View
What is the work
done by the applied force of magnitude
?
Ex p r e s s y o u r a n s w e r i n t e r m s o f s o m e o r a l l o f t h e f o l l o w i n g :
,
,
,
.
ANSWER:
= Correct
Pa r t C
What is the change in the potential energy of the block, distance
, after it has been pushed a
up the incline?
Ex p r e s s y o u r a n s w e r i n t e r m s o f s o m e o r a l l o f t h e f o l l o w i n g :
,
,
,
.
ANSWER:
= All attempts used; correct answer displayed
Now the applied force is changed so that instead of pulling the block up the incline, the force
pulls the block down the incline at a constant speed.
P ar ar t D
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What is the change in potential energy of the block, the incline?
, as it moves a distance
Ex p r e s s y o u r a n s w e r i n t e r m s o f s o m e o r a l l o f t h e f o l l o w i n g :
down
,
,
,
.
,
,
,
.
,
,
,
.
ANSWER:
= Answer Requested
P ar ar t E
What is the work
done by the applied force of magnitude
?
Ex p r e s s y o u r a n s w e r i n t e r m s o f s o m e o r a l l o f t h e f o l l o w i n g :
ANSWER:
=
Correct
Pa r t F
What is the work
done on the block by the frictional force?
Ex p r e s s y o u r a n s w e r i n t e r m s o f s o m e o r a l l o f t h e f o l l o w i n g :
ANSWER:
= Answer Requested
Wo r k o n a S l i d i n g B o x
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A box of mass
is sliding along a horizontal surface.
P ar ar t A
The box leaves position with speed . The box is slowed by a constant frictional force until it comes to rest at position . Find , the the magnitude magnitude of the average frictional force that acts on on the the box. box. (Since you don't know the coefficient of friction, don't include it in your answer.) Hint A.1
H o w t o a p p r o a c h t h e p r o b l em em
Hint not displayed
Hint A.2
Fi n d t h e i n i t i a l k i n e t i c e n e r g y
Hint not displayed
Hint A.3
Fi n d t h e f i n a l k i n e t i c e n e r g y
Hint not displayed
Hint A.4
F i n d t h e w o r k d o n e b y f r ic ic t i o n
Hint not displayed Ex p r e s s t h e f r i c t i o n a l f o r c e i n t e r m s o f
,
, and
.
ANSWER:
= Correct
Pa r t B
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After the box comes to rest at at position position , a person starts pushing the box, giving it a speed . When the box reaches position (where ), how much work has the person done on the box? Assume that the box reaches after the person has accelerated it from rest to speed . Hint B.1
H o w t o a p p r o a c h t h e p r o b l em em
Hint not displayed
Hint B.2
F i n d t h e w o r k d o n e b y f r ic ic t i o n
Hint not displayed
Hint B.3
Fi n d t h e c h a n g e i n k i n e t i c e n e r g y
Hint not displayed Ex p r e s s t h e w o r k i n t e r m s o f
,
,
,
, and
.
ANSWER:
= Correct
P S S 9 . 1 Ro Ro t a t i o n a l E n e r g y L e a r n i n g G o a l : To practice Problem-Solving Strategy 9.1 Rotational Energy.
A frictionless pulley has the shape of a uniform solid disk of mass 4.00
and radius 25.0
. A 1.90 stone is attached to a very light wire that is wrapped around the rim of the pulley, and the stone is released from rest. As it falls down, the wire unwinds without
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stretching or slipping, causing the pulley to rotate. How far must the stone fall so that the pulley has 4.40 of kinetic energy?
P r o b l e m - So So l v i n g S t r a t e g y : R o t a t i o n a l e n e r g y IDENTIFY the relevant concepts : You can use work –energy relationships and conservation of energy to find relationships involving position and motion of a rigid body rotating around a fixed axis. The energy method is usually not helpful for problems that involve elapsed time. SET UP the problem using the following steps : 1. When using the energy energy approach, approach, sketch the initial and final final states of the system. Include the positions and velocities on your sketch. 2. Define your coordinate system, system, particularly the level at which which . You will use it to compute gravitational potential energies. Choose the positive direction to be upward because this is what the equation assumes. 3. Identify all all forces that that do work work that can can ’t be described in terms of potential energy. A free-body diagram is always helpful. 4. List the unknown unknown and known quantities, quantities, including the coordinates coordinates and velocities velocities at each point. Decide which unknowns are your target variables. 5. Many problems problems involve a rope or cable wrapped wrapped around around a rotating rigid body, which which functions as a pulley. In these situations, a point on the pulley that contacts the rope has the same linear speed as the rope, provided the rope doesn ’t slip on the pulley. You can then take advantage of the following equations that relate the linear speed and tangential acceleration of a point on a rigid body to the angular velocity and angular acceleration of the body:
EXECUTE the solution as follows : Write Wri te the exp expres ressio sions ns for the ini initia tiall and fin final al kin kineti etic c and pot poten entia tiall ene energi rgies es ( and
) and the nonconservative work
,
,
(if any). Rotational kinetic energy
is expressed expressed in terms terms of the the body's body's moment of inertia . Substitute these expressions into
,
and its angular angular speed (if nonconserva nonconservative tive
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work is done) or (if only conservative work is done) and solve for the target variable(s). EVALUATE your answer : As always, check whether your answer makes physical sense. I D EN EN T I F Y t h e r e l e v a n t c o n c e p t s
This problem describes a system where a rigid body, the pulley, rotates around a fixed axis while a second body, the stone, moves vertically. Since no information about elapsed time is provided, energy considerations must be used to find relationships involving the position and the motion of the system. To simplify the problem, assume that the wire is massless and it unwinds without stretching or slipping, so that there is no relative motion between the wire and the pulley. Note that the pulley is assumed to be rotating around its fixed axis without friction. S ET ET U P t h e p r o b l e m u s i n g t h e f o l l o w i n g s t e p s
P ar ar t A
What forces do work on this system? Hint A.1
How to approach this problem
Hint not displayed Ch e c k a l l t h a t a p p l y .
ANSWER:
frict ional force normal force
g r a v it i t a t i o n a l f or or c e tension
Correct
There is friction between the cable and the pulley, producing the no-slip conditions. However, frictional forces do no work because no mechanical energy is lost by the cable moving relative to the pulley. Since no other forces besides gravity do work on this system, .
P ar ar t B
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Compare the magnitude of the stone's velocity and the magnitude of the pulley's linear velocity at the point of contact between the wire and pulley. ANSWER:
The magnitude of the velocity of t he stone is less than t h a t o f t h e p o in in t o f c o n t a c t . T h e m a g n i t u d e o f t h e v e lo lo c i t y o f t h e s t o n e i s t h e s a m e a s t h a t o f t h e p oi oi n t o f c o n t a c t . The magnitude of the velocity of t he stone is greater t h a n t h a t o f t h e p o in in t o f c o n t a c t . Correct
Here is a sketch of the initial and final states of the system.
In the initial state, the stone is at rest, initially located at stone is located at is
. In the final state, the
and the stone and pulley are moving. The target variable
.
E XE XEC U T E t h e s o l u t i o n a s f o l l o w s
P ar ar t C
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How far must the stone fall so that the pulley has 4.40 Hint C. C.1 1
of kinetic energy?
H o w t o a p p r o a c h t h e p r o b l em em
Hint not displayed
Hint C. C.2 2
Fi n d t h e i n i t i a l g r a v i t a t i o n a l p o t e n t i a l e n e r g y
Hint not displayed
Hint C. C.3 3
Fi n d t h e t o t a l i n i t i a l k i n e t i c e n e r g y
Hint not displayed
Hint C. C.4 4
Fi n d t h e f i n a l g r a v i t a t i o n a l p o t e n t i a l e n e r g y
Hint not displayed
Hint C. C.5 5
Fi n d t h e t o t a l f i n a l k i n e t i c e n e r g y
Hint not displayed
Hint C. C.6 6
S e t t i n g t h e i n i t i a l e n e r gy gy e q u a l t o t h e f i n a l e n e r g y
Hint not displayed E x p r e s s y o u r a n s w e r n u m e r i c a l l y i n m e t e r s t o t h r e e s i g n if i f i c a n t f i g u r es es .
ANSWER:
= 0.460 Correct
The potential energy that the stone loses is converted into the stone's and the pulley's kinetic energy.
EV A L U A T E y o u r a n s w e r
Pa r t D
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To see if your results are reasonable, you can compare the final velocity of the stone as it falls down unwinding the wire from the pulley, to the velocity the stone would have if falling the same distance while unconnected to the pulley. What is the velocity of an untethered stone after falling 0.460 from rest? Hint D. D.1 1
Using the proper kinemat ic equations
To find the speed of the stone during free fall, set the change in potential energy of the stone to the kinetic energy after falling the given distance . Ex p r e s s y o u r a n s w e r n u m e r i c a l l y i n m e t e r s p e r s e c o n d .
ANSWER:
3.01 = Answer Requested
While solving Part C. you may have calculated the value of the final speed of the stone to be 2.10 . Now you found that, if untethered, the stone would move faster. This is reasonable because when the stone is connected to the pulley through the wire, the change in potential energy of the stone must equal the sum of the change in kinetic energy of the stone and the pulley. When the stone is free to fall on its own, instead, the same change in potential energy of the stone must equal only the change of its kinetic energy. Thus, the change in kinetic ki netic energy of the stone is larger when the stone is in free fall, causing the stone to reach a higher speed than if it were connected to the pulley.
K i n e t i c En e r g y o f a Du m b b e l l This problem illustrates the two contributions to the kinetic energy of an extended object: rotational kinetic energy and translational translational kinetic energy. You are to find the total kinetic energy of a dumbbell of mass when it is rotating with angular speed and its center of mass is moving translationally with speed speed . Denote Denote the dumbbell's moment of
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inertia about its center of mass by . Note that if you approximat approximate e the spheres as point masses of mass each located a distance from the center and ignore the moment of inertia of the connecting rod, then the moment of inertia of the dumbbell is given by , but this fact will not be necessary for this problem.
P ar ar t A
Find the total kinetic energy Hint A.1
of the dumbbell.
H o w t o a p p r o a c h t h e p r o b l em em
Hint not displayed
Hint A.2
Fi n d t h e r o t a t i o n a l k i n e t i c e n e r g y
Hint not displayed
Hint A.3
F i n d t h e t r a n s l at at i o n a l k i n e t i c e n e r g y
Hint not displayed Express your answ er in term s of
,
,
, and
.
ANSWER:
= Correct
P ar ar t B
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The rotational kinetic energy term is often called the kinetic energy in the center of mass, while the translational kinetic energy term is called the kinetic ki netic energy of the center of mass. You found that the total kinetic energy is the sum of the kinetic energy in the center of mass plus the kinetic energy of the center of mass. A similar decomposition exists for angular and linear momentum. There are also related decompositio decompositions ns that work for systems of masses, not just rigid bodies like li ke a dumbbell. It is important to understand the applicability of the formula . Which of the following conditions are necessary for the formula to be valid? Ch e c k a l l t h a t a p p l y .
ANSWER:
The velocity vect or m u s t b e p e r p e nd nd i c u l a r t o t h e axis of rotation. The velocity vect or m u s t b e p e r p e nd n d i c u l a r o r p a r a l le le l t o t h e a x i s o f r o t a t i o n. n. T h e m o m e n t o f i n e r t i a m u s t b e t a k e n a b ou ou t a n a x i s through the cent er of mass.
Correct
K i n e t i c E n e r g y o f a Ro Ro t a t i n g W h e e l A simple wheel has the form of a solid cylinder of radius with a mass uniformly distributed throughout throughout its volume. The wheel is pivoted on a stationary axle through the axis of the cylinder and rotates about the axle at a constant angular speed. The wheel rotates full revolutions in a time interval . P ar ar t A
What is the kinetic energy Hint A.1
of the rotating wheel?
Wh a t i s t h e f o r m u l a f o r r o t a t i o n a l k i n e t i c e n e r g y ?
Hint not displayed
Hint A.2
Moment of inertia of the w heel
Hint not displayed
Hint A.3
Fi n d t h e a n g u l a r v e l o c i t y
Hint not displayed Express your answ er in term s of
,
,
,
and,
.
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ANSWER:
= Correct
R o t a t i o n a l K i n e t i c E n e r g y a n d Co C o n s e r v a t i o n o f En E n e r g y Ra Ra n k i n g Task The five objects objects of various various masses, each denoted , all all have the same radius. They are all rolling at the same speed as they approach a curved incline.
P ar ar t A
Rank the objects based on the maximum height they reach along the curved incline. Hint A.1
U s i n g e n e r g y c o n s e r va va t i o n
Hint not displayed
Hint A.2
Moment of inertia
Hint not displayed R a n k f r o m l a r g e s t t o s m a l l e s t . T o r a n k i t e m s a s e q u i v a le l e n t , o v e r la la p t h e m .
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ANSWER:
View Correct
A R o ll ll i n g H o l l o w Sp h e r e A hollow spherical shell with mass 1.75
rolls without slipping down a slope that makes
an angle angle of of 38.0 38.0 with the horizo horizontal. ntal. P ar ar t A
Find the magnitude of the acceleration Hint A.1
of the center of mass of the spherical shell.
H o w t o a p p r o a c h t h e p r o b l em em
Hint not displayed
Hint A.2
T r a n s l at a t i o n a l m o t i o n i n t h e x d i r e c t i o n
Hint not displayed
Hint A.3
Torque on the spherical shell
Hint not displayed
Hint A.4
Moment of inertia
Hint not displayed
Hint A.5
R e l a t i o n b e t w e e n t h e t r a n s l a t i o n a l a n d a n gu gu l a r a c c e l e r a t i o n s
Hint not displayed T a k e t h e f r e e -f -f a l l a c c e l e r a t i o n t o b e
= 9 .8 .8 0
.
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ANSWER:
= 3.62 Correct
P ar ar t B
Find the magnitude of the frictional force acting on the spherical shell. Hint B.1
H o w t o a p p r o a c h t h e p r o b l em em
Hint not displayed T a k e t h e f r e e -f -f a l l a c c e l e r a t i o n t o b e
ANSWER:
= 9 .8 .8 0
.
= 4.22 Correct
The frictional force keeps the spherical shell stuck to the surface of the slope, so that there is no slipping as it rolls down. If there were no friction, the shell would simply slide down the slope, as a rectangular box might do on an inclined (frictionless) surface.
P ar ar t C
Find the minimum coefficient of friction slipping as it rolls down the slope. Hint C. C.1 1
needed to prevent the spherical shell from
H o w t o a p p r o a c h t h e p r o b l em em
Hint not displayed ANSWER:
= 0.313 Correct
U n w i n d i n g Cy Cy l i n d e r
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A cylinder with moment of inertia about its center of mass, mass , and radius has a string wrapped around it which is tied to the ceiling . The cylinder's vertical position as a function of time is . At time
the cylinder is
released from rest at a height above the ground.
P ar ar t A
The string constrains the rotational and translational motion of the cylinder. What is the relationship between the angular rotation rate and , the velocity of the center of mass of the cylinder? Rememberr that upward motion corresponds to positive linear velocity, and Remembe counterclockwise rotation corresponds corresponds to positive angular velocity. Hint A.1
K e y t o t h e c o n s t r ai a i n ed ed m o t i o n
Hint not displayed
Hint A.2
V e lo l o c i t y o f c o n t a c t p o in in t
Hint not displayed Ex p r e s s
in term s of
and other given quantities.
ANSWER:
= Correct
P ar ar t B
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In similar problems involving rotating bodies, you will often also need the relationship between angular acceleration, , and linear acceleration, . Find in terms of and . ANSWER:
= Correct
P ar ar t C
Suppose that at a certain instant the the velocity of the cylinder is . What is its total kinetic energy, , at that instant? Hint C. C.1 1
Ro t a t i o n a l k i n e t i c e n e r g y
Hint not displayed
Hint C. C.2 2
Ro t a t i o n a l k i n e t i c e n e r g y i n t e r m s o f
Hint not displayed
Hint C. C.3 3
T r a n s la la t i o n a l k i n e t i c e n e r g y
Hint not displayed Express
in te rms of
,
,
, and
.
ANSWER:
= Correct
P ar ar t D
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Find
, the cylinder's vertical velocity when it hits the ground.
Hint D. D.1 1
Initial energy
Hint not displayed
Hint D. D.2 2
Energy conservation
Hint not displayed Express
, in te rms of
,
,
,
, and
.
ANSWER:
= Correct
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