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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 terms of the work-energy theorem. Work-Energy Theorem The work-energy theorem states , is the work done by all forces that act on the object, and where final kinetic energies, respectively. Part A and are the initial and

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 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 Part C

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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 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: , and are the final and initial potential energies, and where nonconservative forces. is the work due only to

Now, we will revisit the falling stone example using the concept of potential energy. Part D

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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 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

Potential Energy Calculations

Learning Goal: To understand the relationship between the force and the potential energy changes associated with that force and to be able to calculate the changes in potential energy as definite integrals. Imagine that a conservative force field is defined in a certain region of space. Does this sound too abstract? Well, think of a gravitational field (the one that makes apples fall down and keeps the planets orbiting) or an electrostatic field existing around any electrically charged object. If a particle is moving in such a field, its change in potential energy does not depend on the particle's path and is determined only by the particle's initial and final positions. Recall that, in general, the component of the net force acting on a particle equals the negative derivative of the potential energy function along the corresponding axis: .

Therefore, the change in potential energy can be found as the integral

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,

where

is the change in potential energy for a particle moving from point 1 to point 2,

is the

net force acting on the particle at a given point of its path, and particle along its path from 1 to 2.

is a small displacement of the

Evaluating such an integral in a general case can be a tedious and lengthy task. However, two circumstances make it easier: 1. Because the result is path-independent, it is always possible to consider the most straightforward way to reach point 2 from point 1. 2. The most common real-world fields are rather simply defined.

In this problem, you will practice calculating the change in potential energy for a particle moving in three common force fields. Note that, in the equations for the forces, in the y direction, and force field. Part A Consider a uniform gravitational field (a fair approximation near the surface of a planet). Find , where and Hint A.1 Relative directions of . is the unit vector in the x direction, is the unit vector

is the unit vector in the radial direction in case of a spherically symmetrical

and Hint not displayed

Express your answer in terms of

, ,

, and

.

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ANSWER: = Part B Consider the force exerted by a spring that obeys Hooke's law. Find , where , and the spring constant Hint B.1 Hint not displayed Express your answer in terms of , ANSWER: = Part C Finally, consider the gravitational force generated by a spherically symmetrical massive object. The magnitude and direction of such a force are given by Newton's law of gravity: , , and . is positive.

where

;

,

, and

are constants; and

. Find .

Hint C.1 Hint not displayed

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Hint C.2 Hint not displayed Express your answer in terms of ANSWER: = , , , , and .

As you can see, the change in potential energy of the particle can be found by integrating the force along the particle's path. However, this method, as we mentioned before, does have an important restriction: It can only be applied to a conservative force field. For conservative forces such as gravity or tension the work done on the particle does not depend on the particle's path, and the potential energy is the function of the particle's position. In case of a nonconservative force--such as a frictional or magnetic force--the potential energy can no longer be defined as a function of the particle's position, and the method that you used in this problem would not be applicable.

Potential Energy Graphs and Motion

Learning Goal: To be able to interpret potential energy diagrams and predict the corresponding motion of a particle. Potential energy diagrams for a particle are useful in predicting the motion of that particle. These diagrams allow one to determine the direction of the force acting on the particle at any point, the points of stable and unstable equilibrium, the particle's kinetic energy, etc. Consider the potential energy diagram shown. The curve represents the value of potential energy

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as a function of the particle's coordinate . The horizontal line above the curve represents the constant value of the total energy of the particle . The total energy is the sum of kinetic ( ) and potential ( ) energies of the particle. The key idea in interpreting the graph can be expressed in the equation

where

is the x component of the net

force as function of the particle's coordinate . Note the negative sign: It means that the x component of the net force is negative when the derivative is positive and vice versa. For instance, if the particle is moving to the right, and its potential energy is increasing, the net force would be pulling the particle to the left. If you are still having trouble visualizing this, consider the following: If a massive particle is increasing its gravitational potential energy (that is, moving upward), the force of gravity is pulling in the opposite direction (that is, downward). If the x component of the net force is zero, the particle is said to be in equilibrium. There are two kinds of equilibrium:

q

q

Stable equilibrium means that small deviations from the equilibrium point create a net force that accelerates the particle back toward the equilibrium point (think of a ball rolling between two hills). Unstable equilibrium means that small deviations from the equilibrium point create a net force that accelerates the particle further away from the equilibrium point (think of a ball on top of a hill).

In answering the following questions, we will assume that there is a single varying force acting on the particle along the x axis. Therefore, we will use the term force instead of the cumbersome x component of the net force. Part A

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The force acting on the particle at point A is __________. Hint A.1 Hint not displayed Hint A.2 Hint not displayed ANSWER: directed to the right directed to the left equal to zero

Consider the graph in the region of point A. If the particle is moving to the right, it would be "climbing the hill," and the force would "pull it down," that is, pull the particle back to the left. Another, more abstract way of thinking about this is to say that the slope of the graph at point A is positive; therefore, the direction of Part B The force acting on the particle at point C is __________. Hint B.1 Hint not displayed Hint B.2 Hint not displayed ANSWER: directed to the right directed to the left equal to zero is negative.

Part C

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The force acting on the particle at point B is __________. Hint C.1 Hint not displayed ANSWER: directed to the right directed to the left equal to zero , and .

The slope of the graph is zero; therefore, the derivative Part D The acceleration of the particle at point B is __________. Hint D.1 Hint not displayed ANSWER: directed to the right directed to the left equal to zero

If the net force is zero, so is the acceleration. The particle is said to be in a state of equilibrium. Part E If the particle is located slightly to the left of point B, its acceleration is __________. Hint E.1 Hint not displayed ANSWER: directed to the right directed to the left equal to zero

Part F

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If the particle is located slightly to the right of point B, its acceleration is __________. Hint F.1 Hint not displayed ANSWER: directed to the right directed to the left equal to zero

As you can see, small deviations from equilibrium at point B cause a force that accelerates the particle further away; hence the particle is in unstable equilibrium. Part G Name all labeled points on the graph corresponding to unstable equilibrium. Hint G.1 Definition of unstable equilibrium Hint not displayed List your choices alphabetically, with no commas or spaces; for instance, if you choose points B, D, and E, type your answer as BDE. ANSWER: BF Part H Name all labeled points on the graph corresponding to stable equilibrium. Hint H.1 Hint not displayed List your choices alphabetically, with no commas or spaces; for instance, if you choose points B, D, and E, type your answer as BDE. ANSWER: DH Part I Name all labeled points on the graph where the acceleration of the particle is zero. Hint I.1 Hint not displayed List your choices alphabetically, with no commas or spaces; for instance, if you choose points B, D, and E, type your answer as BDE. ANSWER: BDFH Your answer, of course, includes the locations of both stable and unstable equilibrium.

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Part J Name all labeled points such that when a particle is released from rest there, it would accelerate to the left. Part J.1 Part not displayed Part J.2 Part not displayed List your choices alphabetically, with no commas or spaces; for instance, if you choose points B, D, and E, type your answer as BDE. ANSWER: AE Part K Consider points A, E, and G. Of these three points, which one corresponds to the greatest magnitude of acceleration of the particle? Hint K.1 Hint not displayed ANSWER: A E G of the particle is known, one can also use the graph of to draw

If the total energy

conclusions about the kinetic energy of the particle since . As a reminder, on this graph, the total energy Part L is shown by the horizontal line.

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What point on the graph corresponds to the maximum kinetic energy of the moving particle? Hint L.1 Hint not displayed ANSWER: D

It makes sense that the kinetic energy of the particle is maximum at one of the (force) equilibrium points. For example, think of a pendulum (which has only one force equilibrium point--at the very bottom). Part M At what point on the graph does the particle have the lowest speed? ANSWER: B

As you can see, many different conclusions can be made about the particle's motion merely by looking at the graph. It is helpful to understand the character of motion qualitatively before you attempt quantitative problems. This problem should prove useful in improving such an understanding.

PSS 11.1: The Sled That Fled

Learning Goal: To practice Problem-Solving Strategy 11.1 for problems involving conservation of energy. A sled is being held at rest on a slope that makes an angle with the horizontal. After the sled is down the slope and then covers the distance along the horizontal released, it slides a distance between the sled and the ground if terrain before stopping. Find the coefficient of kinetic friction that coefficient is constant throughout the trip.

MODEL: Identify which objects are part of the system and which are in the environment. If possible,

choose a system without friction or other dissipative forces. Some problems may need to be subdivided into two or more parts.

VISUALIZE: Draw a before-and-after pictorial representation and an energy bar chart. A free-body

diagram may be helpful if you're going to calculate work, although often the forces are simple enough to be shown on the pictorial representation.

SOLVE: If the system is both isolated and nondissipative, then the mechanical energy is conserved:

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If there are external or dissipative forces, calculate energy equation

and

. Then use the more general

. Kinematics and/or other conservation laws may be needed for some problems.

ASSESS: Check that your result has the correct units, is reasonable, and answers the question.

Identify the objects that should be considered part of the system. Make reasonable simplifying assumptions. Part A Given the situation described in the problem introduction, should you subdivide this problem into parts? And, if so, how many? ANSWER: no yes; into two parts yes; into three parts

It makes sense to divide this problem into two parts: the motion down the slope and the motion along the horizontal. Part B Which assumptions are reasonable to make in this problem? A. The sled can be treated as a particle moving with constant velocity (not necessarily the same velocity in both parts). B. The sled can be treated as a particle moving with constant acceleration (not necessarily the same acceleration in both parts). C. The sled cannot be treated as a particle. D. The air resistance is significant to the problem. E. The air resistance is negligible. F. The amount of thermal energy generated is significant to the problem. G. The amount of thermal energy generated is negligible. H. The force of friction is constant throughout the motion. I. The force of friction is different for different parts of the motion. J. The sled is the only object in the system. K. The sled and slope are both part of the system. Type alphabetically the letters corresponding to the correct answers. Do not use commas. For instance, if you think assumptions A and B are reasonable, enter AB. ANSWER: BEFIK

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Now draw a before-and-after pictorial representation including both parts of the motion. Then sketch an energy bar chart for each part. Part C For each of the three indicated moments in the motion of the sled, identify which of the incomplete energy bar charts shown best represents the energy distribution at that moment. 1. when the sled is released from rest at the top of the slope 2. when the sled reaches the end of the slope 3. when the sled comes to a stop

Write the letters labeling the energy bar charts that correspond to moments 1, 2 and 3, respectively. Do not use commas. ANSWER: CDB Of course, the reason these sets are incomplete is that they do not show the dissipative terms such , , and . You should make sure that your own energy bar charts are complete. as Part D

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In this problem, ANSWER:

because of the presence of which factor? normal force weight force of friction change in direction acceleration

Here is an example of what a before-and-after pictorial representation and bar charts for this problem might look like.

Now use the information and the insights that you have accumulated to construct the necessary

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mathematical expressions and to derive the solution. Part E Find the coefficient of kinetic friction . Use your pictorial representation and the energy bar chart to obtain the necessary equations. Hint E.1 Hint not displayed Hint E.2 Hint not displayed Part E.3 Part not displayed Part E.4 Part not displayed Express the coefficient of kinetic friction in terms of the given quantities. You may or may not use all of them. ANSWER: = As expected, the answer does not depend on the mass of the sled. When you work on a problem on your own, without the computer-provided feedback, only you can assess whether your answer seems right. The following questions will help you practice the skills necessary for such an assessment. Part F A different sled is released from the same height on the same slope; it ends up traveling a greater distance along the horizontal surface than the first sled described in the problem introduction. The coefficient of kinetic friction between the ground and the second sled is constant throughout the for the first sled trip. Call it . Intuitively, what can one conclude about the relationship of and for the second sled? ANSWER:

The masses of both sleds must be known to determine the relationship between and .

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Part G Which of these mathematical expressions have the appropriate dimensions of the coefficient of kinetic friction? A. B. C. D. E.

Type the letters corresponding to correct answers alphabetically. Do not use commas. For instance, if A, B, and D have the appropriate dimensions, enter ABD. ANSWER: ACE The coefficient of friction is, of course, a dimensionless ratio: It has no units.

The Work-Energy Theorem

Learning Goal: To understand the meaning and possible applications of the work-energy theorem. In this problem, you will use your prior knowledge to derive one of the most important relationships in mechanics: the work-energy theorem. We will start with a special case: a particle of mass moving in the x direction at constant acceleration . During a certain interval of time, the particle to , undergoing displacement given by . accelerates from Part A Find the acceleration Hint A.1 of the particle. Hint not displayed Express the acceleration in terms of ANSWER: = , , and .

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Part B Find the net force Hint B.1 Hint not displayed Express your answer in terms of ANSWER: = Part C Find the net work done on the particle by the external forces during the particle's motion. and . and . acting on the particle.

Express your answer in terms of ANSWER: =

Part D Substitute for from Part B in the expression for work from Part C. Then substitute for from the relation in Part A. This will yield an expression for the net work done on the particle by the external forces during the particle's motion in terms of mass and the initial and final velocities. Give an expression for the work in terms of those quantities. Express your answer in terms of ANSWER: = , , and .

The expression that you obtained can be rearranged as

The quantity

has the same units as work. It is called the kinetic energy of the moving . Therefore, we can write

particle and is denoted by

and

.

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Note that like momentum, kinetic energy depends on both the mass and the velocity of the moving object. However, the mathematical expressions for momentum and kinetic energy are different. Also, unlike momentum, kinetic energy is a scalar. That is, it does not depend on the sign (therefore direction) of the velocities. Part E done on the particle by the external forces during the motion of the particle Find the net work in terms of the initial and final kinetic energies. Express your answer in terms of ANSWER: = and .

This result is called the work-energy theorem. It states that the net work done on a particle equals the change in kinetic energy of that particle. Also notice that if is zero, then the work-energy theorem reduces to . In other words, kinetic energy can be understood as the amount of work that is done to accelerate the particle from rest to its final velocity. The work-energy theorem can be most easily used if the object is moving in one dimension and is being acted upon by a constant net force directed along the direction of motion. However, the theorem is valid for more general cases as well. Let us now consider a situation in which the particle is still moving along the x axis, but the net force, which is still directed along the x axis, is no longer constant. Let's see how our earlier definition of work,

needs to be modified by being replaced by an integral. If the path of the particle is divided into very small displacements , we can assume that over each of these small displacement intervals, the net force remains essentially constant and the work done to move the particle from , to is

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where is the x component of the net force (which remains virtually constant for the small displacement from to ). The net work done on the particle is then given by . Now, using

and ,

it can be shown that .

Part F Evaluate the integral Hint F.1 Hint not displayed Express your answer in terms of ANSWER: = , , and . .

The expression that you havejust obtained is equivalent to . Not surprisingly, we are back to the same expression of the work-energy theorem! Let us see how the theorem can be applied to problem solving. Part G

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A particle moving in the x direction is being acted upon by a net force constant . The particle moves from kinetic energy of the particle during that time? Hint G.1 to . What is

, for some , the change in

Hint not displayed Hint G.2 Hint not displayed Express your answer in terms of ANSWER: = It can also be shown that the work-energy theorem is valid for two- and three-dimensional motion and for a varying net force that is not necessarily directed along the instantaneous direction of motion of the particle. In that case, the work done by the net force is given by the line integral and .

where

and

are the initial and the final positions of the particle, is the net force acting on the particle.

is the vector

representing a small displacement, and

Where's the Energy?

Learning Goal: To understand how to apply the law of conservation of energy to situations with and without nonconservative forces acting. The law of conservation of energy states the following: In an isolated system the total energy remains constant. If the objects within the system interact through gravitational and elastic forces only, then the total mechanical energy is conserved. and potential energy . For such systems where no forces other than the gravitational and elastic forces do work, the law of conservation of energy can be written as The mechanical energy of a system is defined as the sum of kinetic energy

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, where the quantities with subscript "i" refer to the "initial" moment and those with subscript "f" refer to the final moment. A wise choice of initial and final moments, which is not always obvious, may significantly simplify the solution. The kinetic energy of an object that has mass and velocity . is given by

Potential energy, instead, has many forms. The two forms that you will be dealing with most often in this chapter are the gravitational and elastic potential energy. Gravitational potential energy is the energy possessed by elevated objects. For small heights, it can be found as , where is the mass of the object, is the acceleration due to gravity, and is the elevation of the object above the zero level. The zero level is the elevation at which the gravitational potential energy is assumed to be (you guessed it) zero. The choice of the zero level is dictated by convenience; typically (but not necessarily), it is selected to coincide with the lowest position of the object during the motion explored in the problem. Elastic potential energy is associated with stretched or compressed elastic objects such as springs. For a spring with a force constant , stretched or compressed a distance , the associated elastic potential energy is .

When all three types of energy change, the law of conservation of energy for an object of mass can be written as .

The gravitational force and the elastic force are two examples of conservative forces. What if nonconservative forces, such as friction, also act within the system? In that case, the total mechanical energy would change. The law of conservation of energy is then written as

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,

where

represents the work done by the nonconservative forces acting on the object between the

initial and the final moments. The work is usually negative; that is, the nonconservative forces tend to decrease, or dissipate, the mechanical energy of the system. In this problem, we will consider the following situation as depicted in the diagram : A block of mass slides at a speed along a horizontal, smooth table. It next slides down a smooth ramp, descending a height , and then slides along a horizontal rough floor, stopping eventually. Assume that the block slides slowly enough so that it does not lose contact with the supporting surfaces (table, ramp, or floor). You will analyze the motion of the block at different moments using the law of conservation of energy. Part A Which word in the statement of this problem allows you to assume that the table is frictionless? ANSWER: straight smooth horizontal

Although there are no truly "frictionless" surfaces, sometimes friction is small enough to be neglected. The word "smooth" often describes such lowfriction surfaces. Can you deduce what the word "rough" means? Part B

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Suppose the potential energy of the block at the table is given by chosen zero level of potential energy is __________. Hint B.1 Definition of Hint not displayed ANSWER: a distance a distance a distance a distance on the floor Part C If the zero level is a distance the floor? Express your answer in terms of some or all the variables constants. ANSWER: = Part D Considering that the potential energy of the block at the table is , what is the change in potential energy to the floor? Hint D.1 Hint not displayed Express your answer in terms of some or all the variables constants. ANSWER: = , , and , , and above the floor below the floor above the floor below the floor

. This implies that the

above the floor, what is the potential energy

of the block on

and any appropriate

and that on the floor is

of the block if it is moved from the table

and any appropriate

As you may have realized, this choice of the zero level was legitimate but not very convenient. Typically, in such problems, the zero level is assumed to be on the floor. In solving this problem, we will assume just that: the zero level of potential energy is on the floor.

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Part E Which form of the law of conservation of energy describes the motion of the block when it slides from the top of the table to the bottom of the ramp? Hint E.1 Hint not displayed ANSWER:

Part F As the block slides down the ramp, what happens to its kinetic energy total mechanical energy ? ANSWER: decreases; decreases; increases; increases; Part G increases; increases; increases; decreases; stays the same increases increases stays the same , potential energy , and

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Using conservation of energy, find the speed Hint G.1

of the block at the bottom of the ramp.

Hint not displayed Express your answer in terms of some or all the variables constants. ANSWER: = Part H Which form of the law of conservation of energy describes the motion of the block as it slides on the floor from the bottom of the ramp to the moment it stops? Hint H.1 Hint not displayed ANSWER: , , and and any appropriate

Part I

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As the block slides across the floor, what happens to its kinetic energy total mechanical energy ? ANSWER: decreases; increases; decreases; increases; decreases; increases; Part J increases; decreases; decreases decreases decreases decreases

, potential energy

, and

stays the same; stays the same; increases; decreases;

stays the same stays the same

What force is responsible for the decrease in the mechanical energy of the block? ANSWER: tension gravity friction normal force

Part K Find the amount of energy Hint K.1 Hint not displayed Express your answer in terms of some or all the variables constants. ANSWER: = , , and and any appropriate dissipated by friction by the time the block stops.

Delivering Rescue Supplies

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You are a member of an alpine rescue team and must project a box of supplies, with mass , up an incline of constant slope angle so that it reaches a stranded skier 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 . Part A Use the work-energy theorem to calculate the minimum speed bottom of the incline so that it will reach the skier. Hint A.1 How to approach the problem Hint not displayed Part A.2 Find the total work done on the box Part not displayed Part A.3 What is the initial kinetic energy? Part not displayed Part A.4 What is the final kinetic energy? Part not displayed Express your answer in terms of some or all of the variables ANSWER: = , , , , and . that you must give the box at the

Energy Required to Lift a Heavy Box

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.

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Use

for the magnitude of the acceleration due to gravity and neglect friction forces.

Part A What is the magnitude with constant velocity? Part A.1 of the upward force you must apply to the rope to start raising the box

Part not displayed Part A.2 Part not displayed Part A.3 Part not displayed Hint A.4 Hint not displayed Express the magnitude of the force in terms of ANSWER: = Part B to a height using two different methods: lifting the box Consider lifting a box of mass directly or lifting the box using a pulley (as in the previous part). , the ratio of the work done lifting the box directly to the work done lifting the What is box with a pulley? Hint B.1 Definition of work Hint not displayed Part B.2 Ratio of the forces , the mass of the box.

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Part not displayed Part B.3 Ratio of the distances Part not displayed Express the ratio numerically. ANSWER: = 1.00

No matter which method you use to lift the box, its gravitational potential energy will increase by . So, neglecting friction, you will always need to do an amount of work equal to to lift it.

Vector Dot Product

Let vectors Calculate the following: Part A Hint A.1 Hint not displayed ANSWER: Part B What is the angle Hint B.1 Hint not displayed ANSWER: Part C ANSWER: = 2.33 between and ? = -10.0 , , and .

= 30.0

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Part D ANSWER: Part E Which of the following can be computed? Hint E.1 Hint not displayed ANSWER:

= 30.0

and Part F Hint F.1

are different vectors with lengths

and

respectively. Find the following:

Hint not displayed Hint F.2 Hint not displayed Express your answer in terms of ANSWER: = Part G

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If

and

are perpendicular,

Hint G.1 Hint not displayed ANSWER: =

Part H If and are parallel,

Hint H.1 Hint not displayed Express your answer in terms of ANSWER: = and .

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 gravity. Hint A.1 Hint not displayed Hint A.2 Hint not displayed Express your answer in kilograms to two significant figures. ANSWER: = 60.0 Part B of the elevator? Use for the magnitude of the acceleration of

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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 Hint not displayed Part B.2 Part not displayed Express your answer in newtons to two significant figures. ANSWER: = 480

The Work Done in Pulling a Supertanker

Two tugboats pull a disabled supertanker. Each tug exerts a constant force of 2×106 , one at an angle 16.0 west of north, and the other at an angle 16.0 east of north, as they pull the tanker a distance 0.650 toward the north. Part A What is the total work done by the two tugboats on the supertanker? Hint A.1 Hint not displayed Part A.2 Part not displayed Express your answer in joules. ANSWER: 2.50×109

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