Assignment 3: Work and Energy Due: 2:00am on Saturday, September 18, 2010 Note: To understand how points are awarded, read your instructor's Grading Policy. [Switch to Standard Assignment View]
Bungee Jumping 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 when unstretched, will first straighten and then stretch as Kate falls.
length
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. Hint A.1
Decide how to approach the problem Hint not displayed
Hint A.2
Compute the force due to the bungee cord Hint not displayed
Express the distance in terms of quantities given in the problem introduction. ANSWER: =
Correct
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. Hint B.1
Decide how to approach the problem Hint not displayed
Hint B.2
Find the initial gravitational potential energy Hint not displayed
Hint B.3
Find the elastic potential energy in the bungee cord
Hint B.3
Find the elastic potential energy in the bungee cord Hint not displayed
Express
in terms of
,
,
, and
.
ANSWER: =
Correct
Dancing Balls Four balls, each of mass
, are connected by four identical relaxed springs with spring constant
balls are simultaneously given equal initial speeds
. The
directed away from the center of symmetry of the
system.
Part A As the balls reach their maximum displacement, their kinetic energy reaches __________. ANSWER:
a maximum zero neither a maximum nor zero Correct
Part B Use geometry to find
, the distance each of the springs has stretched from its equilibrium position. (It
may help to draw the initial and the final states of the system.)
Express your answer in terms of position. ANSWER: =
Correct
, the maximum displacement of each ball from its initial
Correct
Part C Find the maximum displacement Hint C.1
of any one of the balls from its initial position.
A useful equation Hint not displayed
Express
in terms of some or all of the given quantities ,
, and
.
ANSWER: =
Correct
If You Don't Want to Walk to the Kitchen... As depicted in the applet, Albertine finds herself in a very odd contraption. She sits in a reclining chair, in front of a large, compressed spring. The spring is compressed 5.00 from its equilibrium position, and a glass sits 19.8
from her outstretched foot.
Part A For what value of the spring constant
does Albertine just reach the glass without knocking it over?
Determine the answer "experimentally" by playing with the applet.
Express your answer in newtons per meter. ANSWER:
= 95.0 Correct
Part B Assuming that Albertine's mass is 60.0 chair and the waxed floor? Use Assume that the value of
= 9.80
, the coefficient of kinetic friction between the
for the magnitude of the acceleration due to gravity.
found in Part A has three significant figures.
Note that if you did not assume that significant figures for
, what is
has three significant figures, it would be impossible to get three
, since the length scale along the bottom of the applet does not allow you to
measure distances to that accuracy with different values of Hint B.1
.
How to approach the problem Hint not displayed
Hint B.2
Find Albertine's initial potential energy Hint not displayed
Hint B.3
Find the work done by nonconservative forces Hint not displayed
Express your answer to three significant figures. ANSWER:
= 0.102 Correct
Part C The principle of conservation of energy states that energy is neither created nor destroyed. Which of the following describes the transformation of energy in this problem? ANSWER:
Conservation of energy does not apply to problems involving nonconservative forces. Thus, the potential energy slowly disappears during Albertine's trip. The potential energy was turned into Albertine's kinetic energy, which was then converted into internal (thermal) energy. The potential energy was turned into Albertine's kinetic energy, which is now stored in the floor as frictional potential energy. The potential energy was turned into elastic frictional energy, creating the frictional force. Correct
This applet shows how the energy transforms throughout Albertine's journey. Notice that her kinetic energy is never equal to her initial potential energy, because friction is acting even as the spring expands. Try changing the spring constant and observe how the transformation of energy is affected.
Loop the Loop A roller coaster car may be approximated 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.
Part A Find an expression for the kinetic energy of the car at the top of the loop.
Hint A.1
Find the potential energy at the top of the loop Hint not displayed
Express the kinetic energy in terms of ANSWER:
,
,
, and
.
=
Correct
Part B Find the minimum initial height
at which the car can be released that still allows the car to stay in
contact with the track at the top of the loop. Hint B.1
How to approach this part Hint not displayed
Hint B.2
Acceleration at the top of the loop Hint not displayed
Hint B.3
Normal force at the top of the loop Hint not displayed
Hint B.4
Solving for Hint not displayed
Express the minimum height in terms of 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
.
Work Raising an Elevator Look at this applet. It shows an elevator with a small initial upward velocity being raised by a cable. The tension in the cable is constant. The energy bar graphs are marked in intervals of 600 . Part A
What is the mass
of the elevator? Use
for the magnitude of the acceleration of
gravity. Hint A.1
Using the graphs Hint not displayed
Hint A.2
Needed formula Hint not displayed
Express your answer in kilograms to two significant figures. ANSWER:
= 60 Correct
Part B Find the magnitude of the tension
in the cable. Be certain that the method you are using will be
accurate to two significant figures. Hint B.1
How to approach the problem Hint not displayed
Hint B.2
Find the change in mechanical energy Hint not displayed
Express your answer in newtons to two significant figures. ANSWER:
= 480 Correct
Power Dissipation Puts a Drag on Racing 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 with and so does the amount of force you must exert on each small volume of air). Thus , where
is the cross-sectional area of the vehicle and
is called the coefficient of drag.
Part A Consider a vehicle moving with constant velocity Hint A.1
. Find the power dissipated by form drag.
How to approach the problem Hint not displayed
Express your answer in terms of
,
, and speed
.
ANSWER:
=
Correct
Part B A certain car has an engine that provides a maximum power the car,
. Suppose that the maximum speed of
, 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
Find the relationship between speed and power Hint not displayed
Hint B.2
How is the algebra done? Hint not displayed
Express the percent increase in top speed numerically to two significant 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.
Pulling a Block on an Incline with Friction A block of weight
sits on an inclined plane as shown. A force of magnitude
block up the incline at constant speed. The coefficient of kinetic friction between the plane and the block is .
is applied to pull the
Part A What is the total work
done on the block by the force of friction as the block moves a distance
up the incline? Hint A.1
How to start Hint not displayed
Hint A.2
Find the magnitude of the friction force Hint not displayed
Express the work done by friction in terms of any or all of the variables ANSWER:
,
,
,
,
, and
=
Correct
Part B What is the total work
done on the block by the applied force
as the block moves a distance
up the incline?
Express your answer in terms of any or all of the variables ANSWER:
,
,
,
,
, and
.
=
Correct
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.
Part C What is the total work
done on the block by the force of friction as the block moves a distance
down the incline?
Express your answer in terms of any or all of the variables
,
,
,
,
, and
.
.
ANSWER:
=
Correct
Part D What is the total work
done on the box by the appled force in this case?
Express your answer in terms of any or all of the variables ANSWER:
,
,
,
,
, and
.
=
Correct
Dragging a Board A uniform board of length
and mass
lies near a boundary that separates two regions. In region 1,
the coefficient of kinetic friction between the board and the surface is is
, and in region 2, the coefficient
. The positive direction is shown in the figure.
Part 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
The net force of friction Hint not displayed
Hint A.2
Work as integral of force Hint not displayed
Hint A.3
Direction of force of friction Hint not displayed
Hint A.4
Hint A.4
Formula for Hint not displayed
Express the net work in terms of
,
,
,
, and
.
ANSWER: =
Correct This answer makes sense because it is as if the board spent half its time in region 1, and half in region 2, which on average, it in fact did. Part 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 acceleration Hint not displayed
Express your answer in terms of
,
,
,
, and
.
ANSWER: =
Correct
Circling Ball 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 tensions in the string are and . and have magnitudes and .
Part 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.
to that at the top of the circle. Hint A.1
How to approach this problem Hint not displayed
Hint A.2
Find the sum of forces at the bottom of the circle Hint not displayed
Hint A.3
Find the acceleration at the bottom of the circle Hint not displayed
Hint A.4
Find the tension at the bottom of the circle Hint not displayed
Hint A.5
Find the sum of forces at the top of the circle Hint not displayed
Hint A.6
Find the acceleration at the top of the circle Hint not displayed
Hint A.7
Find the tension at the top of the circle Hint not displayed
Hint A.8
Find the relationship between
and
Hint not displayed Express the difference in tension in terms of
and
. The quantities
and
should not
appear in your final answer. ANSWER:
=
Correct The method outlined in the hints is really the only practical way to do this problem. If done properly, finding the difference between the tensions, , can be accomplished fairly simply and elegantly.
Drag on a Skydiver A skydiver of mass velocity of magnitude
jumps from a hot air balloon and falls a distance
before reaching a terminal
. Assume that the magnitude of the acceleration due to gravity is .
Part A What is the work Hint A.1
done on the skydiver, over the distance
How to approach the problem
, by the drag force of the air?
Hint A.1
How to approach the problem Hint not displayed
Hint A.2
Find the change in potential energy Hint not displayed
Hint A.3
Find the change in kinetic energy Hint not displayed
Express the work in terms of
,
,
, and the magnitude of the acceleration due to gravity .
ANSWER: =
Correct
Part B Find the power Hint B.1
supplied by the drag force after the skydiver has reached terminal velocity .
How to approach the problem Hint not displayed
Hint B.2
Magnitude of the drag force Hint not displayed
Hint B.3
Relative direction of the drag force and velocity Hint not displayed
Express the power in terms of quantities given in the problem introduction. ANSWER:
=
Correct
Shooting a ball into a box 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 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
Part A By what distance,
, 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
General method for finding
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 relations for , in terms of
the marble, and solve for its range,
Hint A.2
,
,
, and
.
Initial speed of the marble
Use conservation of energy to find the initial speed,
Express your answer in terms of
,
, and
, of the second marble.
.
ANSWER: =
Correct
Hint A.3
Time for the marble to hit the ground Hint not displayed
Hint A.4
Combining equations and solving for Hint not displayed
Express the distance in terms of
,
,
,
, and
.
ANSWER: =
Correct
Part 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 expression for that depends only on and distance measurements. Hint B.1
Compute Hint not displayed
Hint not displayed Express
in terms of
,
, and
.
ANSWER: =
Correct
Score Summary: Your score on this assignment is 96.9%. You received 38.74 out of a possible total of 40 points.
