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Part A Which of the following statements accurately describes the sign of the work done on the box by the force of the push? Hint A.1 Find the angle Hint not displayed ANSWER: positive negative zero Correct

Part B Which of the following statements accurately decribes the sign of the work done on the box by the normal force? Hint B.1 Finding theta Hint not displayed ANSWER: positive negative zero Correct

Part C Which of the following statements accurately decribes the sign of the work done on the box by the force of kinetic friction? Hint C.1 Finding theta Hint not displayed ANSWER: positive negative

zero Correct

Part D Which of the following statements accurately decribes the sign of the work done on the box by the force of gravity (i.e., the weight)? Hint D.1 Finding the angle Hint not displayed ANSWER: positive negative zero Correct

Making generalizations You may have noticed that the weight and normal forces do no work on the box. Any force that is perpendicular to the displacement of the object on which it acts does no work on the object. The force of kinetic friction did negative work on the box. In other words, it took energy away from the box. Typically, this energy gets transformed into heat, like the heat that radiates from your skin when you get a rug burn due to the friction between your skin and the carpet. A force that acts on an object in a direction opposite to the direction of the object's displacement does negative work on the object. The pushing force acts on the box in the same direction as the object's displacement and does positive work on the box. These generalizations allow physicists to rewrite the equation for work as , where If

is the component of

is parallel to

that is either parallel or antiparallel to the displacement.

, as in the case of

to , as in the case of Part E

, then the work done is positive. If

is antiparallel

, then the work done is negative.

You have just moved into a new apartment and are trying to arrange your bedroom. You

would like to move your dresser of weight 3,500 across the carpet to a spot 5 away on the opposite wall. Hoping to just slide your dresser easily across the floor, you do not empty your clothes out of the drawers before trying to move it. You push with all your might but cannot move the dresser before becoming completely exhausted. How much work do you do on the dresser? ANSWER :

Correct

that to a physicist work means something very specific, and since you were unable to move the dresser, and therefore . However, you got tired and sweaty trying to move the dresser, just as you do when you go to "work out" at the gym.Your muscles are not static strips of fibrous tissue. They continually contract and expand a slight amount when you exert them. Chemical energy from food is being transformed into the energy needed to move your muscles. Work is being done inside your muscles, but work is not being done on the dresser. Part F A box of weight

is sliding down a frictionless plane that is inclined at an angle

above

the horizontal, as shown in the figure

.

What is the work done on the box by the force of gravity if the box moves a distance ? Hint F.1 Finding Theta. Hint not displayed ANSWER:

None of these Correct

The angle given to you in a problem is not always the same angle that you use in the equation for work! Part G The planet Earth travels in a circular orbit at constant speed around the Sun. What is the net work done on the Earth by the gravitational attraction between it and the Sun in one complete orbit? Assume that the mass of the Earth is given by given by , and the Earth-Sun distance is given by Hint G.1 Newton's law of universal gravitation

.

, the mass of the Sun is

Hint not displayed Hint G.2 Circumference of a circle Hint not displayed Hint G.3

Finding the angle Hint not displayed

ANSWER:

None of these. Correct

An object undergoing uniform circular motion experiences a net force that is directed in toward the center of the circle; this net force is called the centripetal force. This force is always perpendicular to the distance the object moves and therefore never does any work on the object. Part H A block of mass

is pushed up against a spring with spring constant

been compressed a distance spring? Hint H.1 Hooke's Law

until the spring has

from equilibrium. What is the work done on the block by the

Hint not displayed ANSWER:

None of these.

Correct

The equation for work presented in this problem requires that the force be constant. Because the force exerted on an object varies with the spring's displacement from equilibrium ( ) you cannot use to find the work done by a spring. In actuality the work done by a spring is given by the equation . Congratulations! Now that you have the basics down and have been exposed to some tricky situations involving the equation for work, you are ready to apply this knowledge to new situations.

Work and Kinetic Energy Two blocks of ice, one four times as heavy as the other, are at rest on a frozen lake. A person pushes each block the same distance . Ignore friction and assume that an equal force exerted on each block. Part A

is

Which of the following statements is true about the kinetic energy of the heavier block after the push? Hint A.1 How to approach the problem Hint not displayed Hint A.2 Find the work done on each block Hint not displayed ANSWER :

It is smaller than the kinetic energy of the lighter block. It is equal to the kinetic energy of the lighter block. It is larger than the kinetic energy of the lighter block. It cannot be determined without knowing the force and the mass of each block. Correct

The work-energy theorem states that the change in kinetic energy of an object equals the net work done on that object. The only force doing work on the blocks is the force from the

person, which is the same in both cases. Since the initial kinetic energy of each block is zero, both blocks have the same final kinetic energy. Part B Compared to the speed of the heavier block, how fast does the light block travel? Hint B.1 How to approach the problem Hint not displayed Hint B.2 Proportional reasoning Hint not displayed ANSWER :

one quarter as fast half as fast the same speed twice as fast four times as fast Correct

Since the kinetic energy of the lighter block is equal to the kinetic energy of the heavier block, the lighter block must be moving faster than the heavier block. Part C Now assume that both blocks have the same speed after being pushed with the same force . What can be said about the distances the two blocks are pushed? Hint C.1 How to approach the problem Hint not displayed Hint C.2 Relate the kinetic energies of the blocks Hint not displayed Hint C.3

Compare the amount of work done on each block Hint not displayed

ANSWER:

The heavy block must be pushed 16 times farther than the light block. The heavy block must be pushed 4 times farther than the light block. The heavy block must be pushed 2 times farther than the light block. The heavy block must be pushed the same distance as the light block. The heavy block must be pushed half as far as the light block. Correct

Because the heavier block has four times the mass of the lighter block, when the two blocks travel with the same speed, the heavier block will have four times as much kinetic energy. The work-energy theorem implies that four times more work must be done on the heavier block than on the lighter block. Since the same force is applied to both blocks, the heavier block must be pushed through four times the distance as the lighter block.

Introduction to Potential Energy Learning Goal: Understand that conservative forces can be removed from the work integral by incorporating them into a new form of energy called potential energy that must be added to the kinetic energy to get the total mechanical energy. The first part of this problem contains short-answer questions that review the work-energy theorem. In the second part we introduce the concept of potential energy. But for now, please answer in of the work-energy theorem. Work-Energy Theorem The work-energy theorem states , where is the work done by all forces that act on the object, and and final kinetic energies, respectively. Part A

and

are the initial

The work-energy theorem states that a force acting on a particle as it moves over a ______ changes the ______ energy of the particle. Choose the best answer to fill in the blanks above:

ANSWER :

distance / potential

distance / kinetic vertical displacement / potential none of the above Correct

Part B To calculate the change in energy, you must know the force as a function of _______. The work done by the force causes the energy change. Choose the best answer to fill in the blank above:

ANSWER :

acceleration work distance potential energy Correct

Part C To illustrate the work-energy concept, consider the case of a stone falling from to under the influence of gravity. Using the work-energy concept, we say that work is done by the gravitational _____, resulting in an increase of the ______ energy of the stone. Choose the best answer to fill in the blanks above:

ANSWER :

force / kinetic potential energy / potential force / potential

potential energy / kinetic Correct

Potential Energy You should read about potential energy in your text before answering the following questions. Potential energy is a concept that builds on the work-energy theorem, enlarging the concept of energy in the most physically useful way. The key aspect that allows for potential energy is the existence of conservative forces, forces for which the work done on an object does not depend on the path of the object, only the initial and final positions of the object. The gravitational force is conservative; the frictional force is not. The change in potential energy is the negative of the work done by conservative forces. Hence considering the initial and final potential energies is equivalent to calculating the work done by the conservative forces. When potential energy is used, it replaces the work done by the associated conservative force. Then only the work due to nonconservative forces needs to be calculated. In summary, when using the concept of potential energy, only nonconservative forces contribute to the work, which now changes the total energy: , where and are the final and initial potential energies, and nonconservative forces.

is the work due only to

Now, we will revisit the falling stone example using the concept of potential energy. Part D Rather than ascribing the increased kinetic energy of the stone to the work of gravity, we now (when using potential energy rather than work-energy) say that the increased kinetic energy comes from the ______ of the _______ energy. Choose the best answer to fill in the blanks above:

ANSWER :

work / potential force / kinetic change / potential Correct

Part E

This process happens in such a way that total mechanical energy, equal to the ______ of the kinetic and potential energies, is _______. Choose the best answer to fill in the blanks above:

ANSWER :

sum / conserved sum / zero sum / not conserved difference / conserved Correct

Hill's Law Conceptual Question Imagine that you're loading a pickup truck with bags of groceries. You notice that the smaller the weight you attempt to lift, the quicker you can lift it. However, you also notice that there is a limit to how quickly you can lift even very small weights, and that above a certain weight, you can no longer lift the weight at all. The detailed relationship between the contraction velocity of a muscle (the speed with which you can lift something) and the weight you are attempting to lift, is known as Hill’s law. Part A Based on this description, which of the following graphs of velocity vs. force is a possible

representation of Hill’s law? Hint A.1 Maximum weight Hint not displayed

ANSWER: A B C D E Correc t

Part B The powerexerted by a muscle is the product of the force exerted and the velocity of contraction. The area of which of these shaded regions represents the power exerted while a

weight is lifted at maximum speed? Hint B.1 How to approach the problem Hint not displayed ANSWER: A B C

D None of the above Correct

The power produced by a muscle is represented by the area of the rectangle formed by the two coordinate axes and the point on the Hill’s law graph representing the weight being lifted. Notice that if you lift a very large weight (near the limit of the maximum force your muscle can produce), the area of this "long and skinny" rectangle can be quite small. If you lift a very small weight, the area of this "tall and skinny" rectangle can also be quite small. However, if you lift a weight near the middle of your weightlifting range, the area of the rectangle, and hence the power produced by your muscle, is a maximum.

Stretching a Spring As illustrated in the figure, a spring with spring constant where

is stretched from

to

is the equilibrium position of the spring.

Part A During which interval is the largest amount of energy required to stretch the spring? Hint A.1 How to approach the problem Hint not displayed ANSWER: From

to

,

From

From

to

to

The energy required is the same in all three intervals. Correct

A graph of the force exerted on the spring versus the displacement of the spring is shown in

the figure. Recall that on a graph of force as a function of position, the work done by the force is represented by the area under the curve. The work done by the hand in the first segment to pull the spring from to to

is represented by a single triangle. The area under the second segment from is three times larger than the first segment, and the area under the third segment

from to is five times larger than in the first segment. So more energy is required to pull the spring through the third segment. Part B A spring is stretched from

to

, where

spring. It is then compressed from to required to stretch or compress the spring?

is the equilibrium position of the . What can be said about the energy

Hint B.1

How to approach the problem Hint not displayed

ANSWER:

More energy is required to stretch the spring than to compress it. The same amount of energy is required to either stretch or compress the spring. Less energy is required to stretch the spring than to compress it.

Correct

The work done to stretch or compress a spring from equilibrium is given by , where

is the distance away from equilibrium that the spring moves. Since

the equation for work, stretching ( ) or compressing ( distance requires the same positive amount of work.

is squared in

) a spring by the same

Part C Now consider two springs A and B that are attached to a wall. Spring A has a spring constant that is four times that of the spring constant of spring B. If the same amount of energy is required to stretch both springs, what can be said about the distance each spring is stretched? Hint C.1 How to approach this problem Hint not displayed Hint C.2 Use proportional reasoning to find a relationship between the springs Hint not displayed ANSWER :

Spring A must stretch 4 times as far as spring B Spring A must stretch 2 times as far as spring B. Spring A must stretch the same distance as spring B.

Spring A must stretch half the distance spring B stretches. Spring A must stretch one-quarter of the distance spring B stretches. Correct

The energy required to stretch a spring is proportional to ,

must be half that of

and to

. If

is four times

, so the energy required is the same for both springs.

Part D Two identical springs are attached to two different masses, greater than

and

, where

is

. The masses lie on a frictionless surface. Both springs are compressed the

same distance, , as shown in the figure. Which of the following statements descibes the energy required to compress spring A and spring B?

ANSWER :

Spring A requires more energy than spring B. Spring A requires the same amount of energy as spring B. Spring A requires less energy than spring B.

Not enough information is provided to answer the question. Correct

Good job; you have realized an important fact. The work done on a spring to compress it a distance is given by . The amount of mass attached to the spring does not affect the work required to stretch or compress the spring.

Projectile Motion and Conservation of Energy Ranking Task Part 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 How to approach the problem Hint not displayed Rank from largest to smallest. To rank items as equivalent, overlap them.

ANSWER :

All the same! Top of Form Bottom of Form

View

Correct

This answer is best understood in of conservation of energy. The initial energy of the ball is independent 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.

Kinetic and Potential Energy of Baseball Graphing Question 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. Hint 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 : positive negative zero Correct

Hint A.2 The shape of the kinetic energy graph The ball's speed decreases linearly from its initial value, which we can denote by , because of the constant acceleration due to gravity. The velocity of the ball can be described by the equation . 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 Draw graph (0,100) (1,36) (2,4) (2.5,0) (3,4) (4,36) : (5,100) Top of Form Bottom of Form

View

Correct

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 Hint not displayed Hint B.2 The shape of the gravitational potential energy graph Hint not displayed

Hint B.3

Using conservation of energy Hint not displayed

ANSWER: Draw graph (0,0) (1,64) (2,96) (2.5,100) (3,96) (4,64) (5,0)

Top of Form Bottom of Form

View Correct

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 Hint not displayed ANSWER: Graph Y=100 Top of Form Bottom of Form

View

Correct

Properties of Circular Orbits

Learning Goal: To find some of the parameters characterizing an object moving in a circular orbit. The motivation for Isaac Newton to discover his laws of motion was to explain the properties of planetary orbits that were observed by Tycho Brahe and analyzed by Johannes Kepler. A good starting point for understanding this (as well as the speed of the space shuttle and the height of geostationary satellites) is the simplest orbit: a circular one. This problem concerns the properties of circular orbits for a satellite orbiting a planet of mass For all parts of this problem, where appropriate, use constant. Part A Find the orbital speed

.

for the universal gravitational

of a satellite in a circular orbit of radius

around a planet of

mass . Hint A.1 Find the force Hint not displayed Hint A.2 Find the radial acceleration Hint not displayed Hint A.3 Newton's second law Hint not displayed Express the orbital speed in of

,

, and

.

ANSWER : =

Cor rect

Part B Find the kinetic energy planet of mass

of a satellite with mass

.

Express your answer in of

,

,

, and

.

in a circular orbit of radius

around a

ANSWER : =

Cor rect

Part C The potential energy object of mass

of an object of mass

that is separated by a distance

from an

is given by .

What is the kinetic energy

of the satellite?

Express your answer in of the potential energy

.

ANSWER : =

Cor rec t

You have found that . The total mechanical energy the kinetic and potential energies. This means that

of the satellite is the sum of

. Part D Find the satellite's orbital period . Hint D.1 How to approach the problem Hint not displayed Hint D.2 Find the distance the satellite travels during one orbit Hint not displayed

Express your answer in of

,

,

, and

.

ANSWER : =

Corr ect

Part E Find an expression for the square of the orbital period. Express your answer in of

ANSWER : =

Cor rect

,

,

, and

.