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Course TUPH1061F07
Homework Ch. 7 Due 4 Oct.
Due at 12:00pm on Wednesday, October 10, 2007
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Details
Bungee Jumping
Description: Determine the spring constant of a bungee cord using Newtons 2nd Law. Then compute length of cord needed
to avoid hitting the river below using conservation of energy.
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, will first straighten and then
stretch as Kate falls.
[ Print ]
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 acceleration due to gravity.
Part A
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.
Part A.1
Decide how to approach the problem
the forcefor
due
to thethis
bungee
cord
Part
Here
areA.2
threeCompute
possible methods
solving
problem:
When Kate is at rest, what is the magnitude
of the upward force the bungee cord exerts on her?
a. No nonconservative forces are acting, so mechanical 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 (which depends on the cord's final length
Part A.2.a
of the bungee cord
) and Find
solvethe
for extension
.
The
force on Kate is forces
due to are
the acting,
extension
of the bungee
cord.
What
is this extension?
b. upward
Since nonconservative
mechanical
energy
is not
conserved.
Set the spring potential energy in the
bungee cord (which depends on ) equal to Kate's gravitational potential energy plus the work done by dissipative
Express your answer in terms of the cord's final (stretched) length and .
forces. Eliminate the unknown work, and solve for .
c. When KateExtension
comes to =rest she has zero acceleration, so the net force acting on her must be zero. Set the spring force
ANSWER:
due to the bungee cord (which depends on ) equal to the force of gravity and solve for .
Hint A.2.b Formula for the force due to a stretched cord
The formula
foroptions
the force
duesimplest,
to a stretched
cord is way to find
Which
of these
is the
most accurate
given the information available?
,
ANSWER:
a
b
c
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where
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is the spring constant of the cord and
is the extension of the cord.
Express your answer in terms of the cord's final stretched length
Your answer should not depend on Kate's mass
ANSWER:
and quantities given in the problem introduction.
.
=
Set this force equal to Kate's weight, and solve for .
Express the distance in terms of quantities given in the problem introduction.
ANSWER:
=
Part 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 ? Ignore all dissipative forces.
Part B.1
Decide how to approach the problem
Here are three possible methods for solving this problem:
a. Since nonconservative forces are ignored, mechanical 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. Nonconservative forces can be ignored, so mechanical energy is conserved. Set the spring potential energy in the
bungee cord (which depends on ) equal to Kate's gravitational potential energy at the top of the bridge plus the work
done by gravity as Kate falls. Compute the work done by gravity, then solve for .
c. When Kate is being held just above the water she has zero 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
ANSWER:
a
b
given the information available?
c
Part B.2 Find the initial gravitational potential energy
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.)
Express your answer in terms of quantities given in the problem introduction.
ANSWER:
=
Part B.3 Find the elastic potential energy in the bungee cord
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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.a Formula for elastic potential energy
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.
Part B.3.b How much is the bungee cord stretched?
By how much is the bungee cord stretched when Kate is at a depth
Express your answer in terms of
ANSWER:
below the bridge?
and .
=
Express your answer in terms of quantities given in the problem introduction.
ANSWER:
Express
=
in terms of , ,
ANSWER:
, and .
=
Energy Required to Lift a Heavy Box
Description: Find force needed to lift a box with a single pulley. Then compare energy needed to lift box directly, versus
with a pulley.
As you are trying to move a heavy box of mass , you realize that it is too heavy for you to lift by yourself. There is no one
around to help, so you attach an ideal pulley to the box and a massless rope to the ceiling, which you wrap around the pulley.
You pull up on the rope to lift the box.
Use for the magnitude of the acceleration due to gravity and neglect
friction forces.
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Part A
What is the magnitude
Part A.1
of the upward force you must apply to the rope to start raising the box with constant velocity?
What force must be applied to the box to keep it moving at a constant speed?
What is the magnitude of the force that the pulley must exert on the box so that it moves at a constant speed?
Express your answer in terms of the mass of the box.
ANSWER:
Part A.2
=
What force does the pulley exert on the box?
If you take the tension in the rope to be , what is
, the magnitude of the net upward force that the pulley exerts on the
box?
Express your answer in terms of
ANSWER:
Part A.3
.
=
Find the tension in the rope
Find the tension in the rope in terms of
ANSWER:
Hint A.4
, the force with which you are pulling upward.
=
Putting it all together
On your own or using the previous hints, you should have found equations for he following:
I. the force needed to lift the box in terms of its mass,
II. the relationship between the force on the box due to the pulley and the tension in the rope, and
III. the relationship between the force applied to the rope and the tension in the rope.
Use two of these equations to eliminate the force applied by the pulley and the tension in the rope. You should then be able
to express the force applied on the rope in terms of the mass of the box.
Express the magnitude of the force in terms of
, the mass of the box.
ANSWER:
=
Part B
Consider lifting a box of mass
to a height
using two different methods: lifting the box directly or lifting the box using a
pulley (as in the previous part).
What is
Hint B.1
, the ratio of the work done lifting the box directly to the work done lifting the box with a pulley?
Definition of work
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In each case, the amount of work
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you do is equal to the force
you apply times the distance
over which you apply the
force:
.
Part B.2
Ratio of the forces
What is the ratio of the force needed to lift the box directly to the force needed to lift the box using the pulley?
Express your answer numerically.
ANSWER:
=
2
Part B.3
Ratio of the distances
What is the ratio of the distance over which force is applied when lifting the box directly to the distance over which force is
applied when lifting the box with the pulley?
Part B.3.a Find the distance when using the pulley
Find
, the distance over which you must apply force when lifting the box using the pulley.
Hint B.3.a.i Pulling the rope a short distance
Using the pully, imagine that you pull the end of the rope a short distance
distance
upward. The box will actually rise a
. (Draw a picture if you have trouble visualizing this.)
Express your answer in terms of , the total height that the box is lifted.
ANSWER:
=
Hint B.3.b Find the distance when lifting directly
When lifting the box directly, the distance over which force is applied,
, is equal to the vertical distance
that the box
is raised.
Express the ratio of distances numerically.
ANSWER:
=
0.500
Express the ratio numerically.
ANSWER:
=
1
No matter which method you use to lift the box, its gravitational potential energy will increase by
friction, you will always need to do an amount of work equal to
. So, neglecting
to lift it.
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Fun with a Spring Gun
Description: A ball is launched vertically from a spring gun. Use conservation of energy to compute the velocity of the ball
as a function of height, and to compute the maximum height reached by the ball.
A spring-loaded toy gun is used to shoot a ball of mass
has spring constant
. If the spring is compressed a distance of
25.0 centimeters from its equilibrium position
ball reaches a maximum height
straight up in the air, as shown in the figure. The spring
and then released, the
(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 up
and down along the y axis.
Part A
Which of the following statements are true?
A. Mechanical energy is conserved because no dissipative forces perform work on the ball.
B. The forces of gravity and the spring have potential energies associated with them.
C. No conservative forces act in this problem after the ball is released from the spring gun.
Hint A.1
Nonconservative forces
Dissipative, or nonconservative, forces are those that always oppose the motion of the object on which they act. Forces such
as friction and drag are dissipative forces.
Hint A.2
Forces acting on the ball
The ball is acted on by the spring force only when the two are in contact. The force of tension in the spring is a conservative
force. Also, the ball is always acted on by gravity, which is also a conservative, or nondissipative, force.
Enter the letter(s) of the correct statements in alphabetical order. For example, if A and C are correct, enter AC.
ANSWER:
AB
Part B
Find
the muzzle velocity of the ball (i.e., the velocity of the ball at the spring's equilibrium position
Part B.1
).
Determine how to approach the problem
What physical relationship can you use to solve this problem? Choose the best answer.
ANSWER:
kinematics equations
momentum
Newton's second law
law of conservation of energy
conservation of
Note that the law of conservation of energy applies to closed systems. In this case, such a closed system consists of the
ball and the spring (and, technically, the Earth, but we will follow the traditional, somewhat imprecise, language and
will assume that it is the ball that has gravitaitonal potential energy, not the system "ball-Earth.")
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Hint B.2
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Energy equations
Recall that kinetic energy is given by the equation
,
where
is the speed of the object and
is the object's mass.
Gravitational potential energy is given by
,
where
is the object's height measured from
.
The elastic potential energy of a spring is given by
,
where
is the spring constant and
Part B.3
is the spring's displacement from equilibrium.
Determine which two locations you should examine
Pick the two points along the ball's path that would be most useful to compare in order to find the solution to this problem.
Choose from among the following three points:
A.
B.
, the location of the ball when the spring is compressed.
, the equilibrium position of the spring.
C.
, the maximum height that the ball reaches above the point
.
Enter the letters that correspond to the two points in alphabetical order.
ANSWER:
AB
Because you do not know enough information about the ball at
to the energy at
Part B.4
to find
, you need to compare the energy at
.
Find the initial energy of the system
A useful statement of mechanical energy conservation relating the initial and final kinetic ( ) and potential ( ) energies is
.
In this situation, the initial position is
and the final position is
, which is the equilibrium position of the
spring. What kind(s) of energy does the system "spring-ball" have at the initial position?
ANSWER:
kinetic only elastic potential only gravitational potential only
kinetic and elastic potential
elastic and gravitational potentials
Keep in mind that
kinetic and gravitational potential
at the equilibrium position of the spring. The inital position defined at
will have
negative gravitational potential energy.
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Part B.5
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Determine the final energy
A useful statement of mechanical energy conservation relating the initial and final kinetic ( ) and potential ( ) energies is
.
In this situation, the initial position is
and the final position is
, which is the equilibrium position of the
spring. What kind(s) of energy does the system "spring-ball" have at the final position?
ANSWER:
Hint B.6
kinetic only elastic potential only gravitational potential only
kinetic and elastic potential
elastic and gravitational potentials
kinetic and gravitational potential
Creating an equation
From the hints you now know what kinds of energy are present at the initial and final positions chosen for the ball in this
part of the problem. You also know that
.
It has been determined that
is zero and
potential energy. In addition,
is zero.
ANSWER:
consists of two terms: gravitational potential energy and elastic
=
Part C
Find the maximum height
Part C.1
of the ball.
Choose two locations to examine
Pick the two points along the ball's movement that would be most useful to compare in order to find a solution to this
problem. Choose from among the following three points:
A.
B.
, the location of the ball when the spring is compressed.
, the equilibrium position of the spring.
C.
, the maximum height that the ball reaches measured from
.
Enter the letters that correspond to the two points in alphabetical order.
ANSWER:
AC
BC
You could compare
comparison because using
to either
or
. It is probably most convenient to use
for
requires that you know the energy at the equilibrium position of the spring. Of course,
you do know it, as long as you got that part of the problem correct. For the remainder of the problem, we will use
and
.
Part C.2
Find the initial energy
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A useful statement of mechanical energy conservation is
.
Recall that in the problem statement,
is set to correspond to the equilibrium position of the spring. Therefore, in this
situation, the initial location is at
and the final position should be taken as
.
What kind(s) of energy does the ball have at the initial location?
ANSWER:
Part C.3
kinetic only elastic potential only gravitational potential only
kinetic and elastic potential
elastic and gravitational potentials
kinetic and gravitational potential
Determine the final energy
A useful statement of mechanical energy conservation is
.
In this situation, the initial location is at
energy does the ball have at
, and the final position should be taken as
. What kind(s) of
?
Part C.3.a Find the speed of the ball at the top of its trajectory
What is the speed
of the ball at the top of its trajectory?
Hint C.3.a.i Motion in the vertical direction
Recall from kinematics that a ball travels upward until its speed decreases to zero, at which point it starts falling back to
Earth.
Express your answer numerically, in meters per second.
ANSWER:
=
Recall that
ANSWER:
Hint C.4
. Because the ball has zero veolicty at the peak of its trajectory, it has no kinetic energy.
kinetic only elastic potential only gravitational potential only
kinetic and elastic potential elastic and gravitational potentials
kinetic and gravitational potential
Creating an equation
From the above hints, you now know what kind of energy is present at the inital and final positions chosen for the ball in
this part of the problem. You know that
.
It was determined that
potential energy. In addition,
is zero and that
consists of two terms: gravitational potential energy and elastic
is zero.
Express youramswer numerically, in meters.
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ANSWER:
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=
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.
Part D
Which of the following actions, if done independently, would increase the maximum height reached by the ball?
A. reducing the spring constant
B. increasing the spring constant
C.
D.
E.
F.
G.
decreasing the distance the spring is compressed
increasing the distance the spring is compressed
decreasing the mass of the ball
increasing the mass of the ball
tilting the spring gun so that it is at an angle
degrees from the horizontal
Enter the letters that correspond to the correct answer(s) in alphabetical order.
ANSWER:
BDE
Spring and Projectile
Description: A spring gun is used to launch a ball off of a table with a ramp at the edge. Problem contains a number of
multiple choice questions about changes in kinetic and potential energy (including an energy diagram), then asks for analytic
expressions for the speed of the ball when it leaves the ramp and when it hits the floor.
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, 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.
Part A
Relative to the initial configuration (with the spring relaxed), when the spring has been compressed, the ball-spring system
has
ANSWER:
gained kinetic energy
gained potential energy
lost kinetic energy
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Part B
As the spring expands (after the ball is released) the ball-spring system
ANSWER:
gains kinetic energy and loses potential energy
gains kinetic energy and gains potential energy
loses kinetic energy and gains potential energy
loses kinetic energy and loses potential energy
Part C
As the ball goes up the ramp, it
ANSWER:
gains kinetic energy and loses potential energy
gains kinetic energy and gains potential energy
loses kinetic energy and gains potential energy
loses kinetic energy and loses potential energy
Part D
As the ball falls to the floor (after having reached its maximum height), it
ANSWER:
gains kinetic energy and loses potential energy
gains kinetic energy and gains potential energy
loses kinetic energy and gains potential energy
loses kinetic energy and loses potential energy
Part 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. 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
Part F
Calculate
, the speed of the ball when it leaves the launching ramp.
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Hint F.1
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General approach
Find an expression for the mechanical energy (kinetic plus potential) of the spring and ball when the spring is compressed.
Then find an expression for the mechanical energy of the ball when it leaves the launching ramp. ( will be an unknown in
this expression.) Since energy is conserved, you can set these two expressions equal to each other, and solve for .
Part F.2
Find the initial mechanical energy
Find the total mechanical energy of the ball-spring system when the spring is fully compressed. Take the gravitational
potential energy to be zero at the floor.
Hint F.2.a What contributes to the mechanical energy?
The total initial mechanical energy is the sum of the potential energy of the spring, the gravitational potential energy, plus
any initial kinetic energy of the ball.
ANSWER:
Part F.3
=
Find the mechanical energy at the end of the ramp
Find the total mechanical energy of the ball when it leaves the launching ramp. (At this point, assume that the spring is
relaxed and has no stored potential energy.) Again, take the gravitational potential energy to be zero at the floor.
Express your answer in terms of
ANSWER:
Hint F.4
and other given quantities.
=
Is energy conserved?
Because no nonconservative forces act on the system, energy is conserved:
Express the speed of the ball in terms of , ,
, , , and/or
.
ANSWER:
=
Part G
With what speed will the ball hit the floor?
Hint G.1
General approach
Find an expression for the initial mechanical energy (kinetic plus potential) of the spring and ball. Then find an expression
for the mechanical energy of the ball when it hits the floor. ( will be an unknown in this expression.) Since energy is
conserved, you can set these two expressions equal to each other, and solve for .
Hint G.2
Initial mechanical energy
For the initial mechanical energy, you can use either the expression you found for the mechanical energy of the ball at the
top of the ramp or that for the total mechanical energy of the ball plus spring just before the ball was launched. These two
expressions are equal.
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Part G.3
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Find the final mechanical energy
Find the total mechanincal energy
of the ball when it hits the floor.
Express your answer in terms of
and other given quantities.
ANSWER:
Hint G.4
=
Is energy conserved?
Only conservative forces (gravity, spring) are acting on the ball, so energy is conserved:
Express the speed in terms of , ,
, , , and/or
.
.
ANSWER:
=
Energy in a Spring Graphing Question
Description: Draw kinetic, elastic potential, and total energy plots of a mass and spring system. (graphing applet)
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.
Part 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
Part B
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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 given in the graphing
window).
Part B.1
Determine the sign of the initial elastic potential energy
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:
Part B.2
positive
negative
zero
Determine the sign of the initial kinetic energy
Is the initial kinetic energy of the cart positive, negative, or zero?
ANSWER:
Part B.3
positive
negative
zero
Determine the sign of the final elastic potential energy
When the car reaches the equilibrium position of the spring, is the elastic potential energy positive, negative, or zero?
ANSWER:
Hint B.4
positive
negative
zero
The shape of the elastic potential energy graph
The elastic potential energy of a spring with spring constant
that is stretched or compressed to position
is given by
,
where
is the equilibrium position of the spring.
ANSWER:
View
Part 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:
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View
Kinetic and Potential Energy of Baseball Graphing Question
Description: Graph kinetic, potential, and total energy over time of a vertically thrown baseball. (graphing applet)
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 energy is located at the position at which the ball leaves
the thrower's hand.
Part A
Sketch a graph of the kinetic energy of the baseball.
Part A.1
Determine the sign of the initial kinetic energy
At the instant the ball leaves the thrower's hand, is its kinetic energy positive, negative, or zero?
ANSWER:
Hint A.2
positive
negative
zero
The shape of the kinetic energy graph
The ball's speed decreases linearly from its initial value, which we can denote by
due to gravity. The velocity of the ball can be described by the equation
, because of the constant acceleration
.
Since kinetic energy depends on the square of velocity, how does the kinetic energy vary with time?
Also, note that the ball reaches its maximum height halfway between the time that it leaves the thrower's hand and the
moment it is caught. What is the speed of the ball when it reaches the maximum height?
ANSWER:
View
Part 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
Initial gravitational potential energy
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The point at which the ball leaves the thrower's hand is defined to be the origin of the y axis, and the gravitational potential
energy of the ball depends on the ball's height above the origin.
Hint B.2
The shape of the gravitational potential energy graph
The potential energy of the ball is proportional to its height, and the height of the ball can be described by the equation
.
Hint B.3
Using conservation of energy
Since there are no nonconservative forces acting on the ball, the total energy must remain the same throughout the motion.
Therefore, your graph of potential energy should be shaped such that potential energy plus kinetic energy does not change
during the motion.
ANSWER:
View
Part C
Based on the kinetic and potential energy graphs given, sketch a graph of the baseball's total energy.
Hint C.1
Total energy
The total energy of the baseball is the sum of its kinetic energy and gravitational potential energy.
ANSWER:
View
A Mass-Spring System with Recoil and Friction
Description: A object moving on a surface with friction compresses a spring, then recoils back and stops at exactly the
spring's equilibrium position. Use conservation of energy, explicitly including energy dissipated by friction, to find the spring
constant.
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
on its return trip, it stops.
Part A
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Find , the spring constant.
Part A.1
Why does the object stop?
Why does the object come to rest when it returns to
?
Although more than one answer may be true of the system, you must choose the answer that explains why the object
ultimately comes to a stop.
ANSWER:
When the object reaches
and extension of the spring.
When the object reaches
the second time all of its initial energy has gone into the compression
the second time all of its initial energy has been dissipated by friction.
is an equilibrium position and at this point the spring exerts no force on the object.
At
the force of friction exactly balances the force exerted by the spring on the object.
Part A.2 How does friction affect the system?
Indicate whether the following statements regarding friction are true or false.
A. Work done by friction is equal to
, where
is the mass of an object,
is the magnitude of the acceleration
due to gravity, is the coefficient of kinetic friction, and is the distance the object has traveled.
, where is the coefficient of friction, is the acceleration due to
B. Energy dissipated by friction is equal to
gravity,
is the mass of the object, and is the amount of time (since encountering the spring) the object has been
moving.
C. Friction is a conservative force.
D. Work done by friction is exactly equal to the negative of the energy dissipated by friction.
Enter the letters of the correct statements in alphabetical order. For example, if statements A and C are correct,
enter AC.
ANSWER:
Hint A.3
AD
Energy stored in a spring
The potential energy stored in a spring having constant
that is compressed a distance
is
.
Part A.4 Compute the compression of the spring
By what distance
does the object compress the spring?
Hint A.4.a How to approach this question
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
.
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Hint A.4.b The value of
In its final position, the object is not moving. Also the spring is not compressed. Therefore
.
Part A.4.c Find
What is the value of
?
Hint A.4.c.i How to approach this part
Initially the spring is uncompressed, so the only contribution to the system's energy comes from the kinetic energy of the
object.
Express your answer in terms of some or all of the variables
ANSWER:
, , , and
and , the acceleration due to gravity.
=
Part A.4.d Find
What is the value of
?
Hint A.4.d.i How to approach this part
The only nonconservative force in the system is the frictional force between the object and the surface it's on. If the
object moves through a distance , the work done by friction
is
Express your answer in terms of some or all of the variables
ANSWER:
Express
, , , and
and , the acceleration due to gravity.
=
in terms of , , and .
ANSWER:
=
Hint A.5 Putting it all together
In the previous part, at the two ends of the motion considered, the spring had no energy, so
However, you were able to find a relation for
was not part of the equation.
in terms of the known quantities. To obtain an equation involving , 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.
Hint A.6 The value of
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The value of
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is again zero.
Part A.7 Find
for this part of the motion
What is the value of
for this part of the motion?
Hint A.7.a How to approach this part
Since the spring is at its maximum compression, the object must be momentarily at rest. So the only contribution to the
energy is from the potential energy of the spring.
Express your answer in terms of
ANSWER:
and , the spring constant, so that you end up with an equation containing .
=
Part A.8 Find
for this part of the motion
What is the value of
for this part of the motion?
Hint A.8.a How to approach this part
The only nonconservative force in the system is the frictional force between the object and the surface it's on. If the object
moves through a distance , the work done by friction
is
.
Express your answer in terms of
ANSWER:
Express
, , , and , the acceleration due to gravity.
=
in terms of ,
, , and .
ANSWER:
=
Summary
2 of 7 items complete (28.57% avg. score)
10 of 35 points
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