Answer: ACDHIKNO

a. TRUE - Work is a form of energy, and in fact it has units of energy.

b. FALSE - Watt is the standard metric unit of power; Joule is the standard metric unit of energy.

c. TRUE - A N•m is equal to a Joule.

d. TRUE - A kg•m²/s² is a mass unit times a speed squared unit, making it a kinetic energy unit and equivalent to a Joule.

e. FALSE - Work is not dependent on how rapidly the force displaces an object; power is time-based and calculated by force multiplied by speed.

f. FALSE - Since Superman does not cause a displacement, no work is done; he is merely holding the car to prevent its descent down the hill.

g. FALSE - The upward force does not cause the horizontal displacement so this is a NON-example of work.

h. TRUE - There is a component of force in the direction of displacement and so this is an example of work.

i. TRUE - There is a force and a displacement; the force acts in the opposite direction as the displacement and so this force does negative work.

j. FALSE - For uniform circular motion, the force acts perpendicular to the direction of the motion and so the force never does any work upon the object.

k. FALSE - This is clearly work - a force is causing an object to be displaced.

l. FALSE - If a force acts at a 90-degree angle to the direction of motion, then the force does not do any work at all. Negative work is done when there is a component of force opposite the direction of motion.

m. FALSE - There are many instances in which an individual force does positive work and yet the object maintains a constant speed. Consider a force applied to lift an object at constant speed. The force does positive work. Consider a car moving at constant speed along a level surface. The force of the road on the tires does positive work while air resistance does and equal amount of negative work.

n. TRUE - A force which acts in a direction opposite the motion of an object will do negative work.

o. TRUE - When non-conservative forces do work upon an object, the object will either gain or lose mechanical energy. Mechanical energy is conserved (neither gained nor lost) only when conservative forces do work upon objects.

Answer: ABDEI

a. TRUE - Power is a rate quantity and thus time-based.

b. TRUE - This is the definition of power.

c. FALSE - This is not always the case. A machine can do a lot of work but if it fails to do it rapidly, then it is not necessarily powerful. In fact two machines can do the same task (and therefore the same work), yet they can have drastically different power ratings.

d. TRUE - An equation for computing work in constant speed situations is P=F•v.

e. TRUE - Watt is the unit of power? Yes!!

f. FALSE - Vice versa. If two people do the same job, then they're doing the same amount of work. The person who does it fastest generates more power.

g. FALSE - A N•m is a Joule and that is a unit of work (not power). Think force (N) times distance (m); that's work (J).

h. FALSE - The work would be (m•g)•d or approximately 1200 J. The power is work divided by time - 1200 J/1.5 s = 800 W.

i. TRUE - Since force and speed are given, use Power = F•v. The calculation yields 450 W.

Answer: See questions above; explanations given below.

a. The forward motion is do to the forward pushing; if the force and motion are in the same direction, then the angle is 0 degrees.

b. Friction opposes motion and as such does negative work; the angle is 180 degrees.

c. The force is vertical and the displacement if horizontal; they make a 90 degree angle.

d. Air resistance opposes motion and as such does negative work; the angle is 180 degrees.

e. Friction opposes motion and as such does negative work; the angle is 180 degrees.

f. The frosh applies an upward force to cause an upward displacement; the angle is 0 degrees.

g. For uniform circular motion, the force is inwards and the displacement at each instant is tangent to the circle; these two vectors make a 90 degree angle.

h. This is a straightforward question; no tricks here.

i. The forward motion is do to the forward pushing; if the force and motion are in the same direction, then the angle is 0 degrees.

j. The cable pulls up on the elevator and the elevator is displaced upward; if the force and motion are in the same direction, then the angle is 0 degrees.

k. The 30-degree angle is the incline angle, not necessarily the angle between F and d. The force is parallel to the incline and the cart is displaced along the direction of the incline; so the two vectors are in the same direction and the angle between them is 0 degrees.

l. Compare the wording of this to part h. This one is tricky because the angle between F and d is 60-degrees. If you missed it, reread the question, paying careful attention to the "with the vertical" part.

m. As the child swings, she traces out a circular arc and as such the tension (centripetal) is perpendicular to the direction of motion (tangent).

a. Positive Work

b. Negative Work

c. No Work

Answer: See table above; explanations provided below.

a. The force is upwards and the displacement is upwards. When the force and the displacement act in the same direction, positive work is done.

b. The force is horizontal and the displacement is horizontal. When the force and the displacement act in the same direction, positive work is done. (It is true that the wall is doing negative work upon the nail but this statement is about the hammer's force on the nail.)

c. The force is backwards and the displacement is forwards. When the force and the displacement act in the opposite direction, negative work is done.

d. The force is backwards and the displacement is forwards. When the force and the displacement act in the opposite direction, negative work is done.

e. If the force does not cause the object to be displaced (the object hangs motionless), then no work is done.

f. The force is backwards and the displacement is forwards. When the force and the displacement act in the opposite direction, negative work is done.

g. The force is upwards and parallel to the incline and the displacement is in the same direction parallel to the incline. When the force and the displacement act in the same direction, positive work is done.

h. As the bob swings downwards from its highest position, the motion is downwards (and rightwards); the force is also downwards and as such there is a component of force in the direction of motion. When the force and the displacement act in the same direction, positive work is done. (Note that if the bob was swinging upwards from its lowest position to its highest position, then gravity would be doing negative work.)

Answer: A(sort of) CDGH I(sort of) J

a. TRUE (sort of) - If a force does work, yet does not remove mechanical energy from an object, then it is definitely a conservative force. The sort of indicates that a force is also considered a conservative force if it does work and does not add mechanical energy to an object.

b. FALSE - If a force does not add mechanical energy to a system of objects, then it is likely a conservative force (provided it doesn't remove mechanical energy either). Non-conservative forces are those which either add or remove energy from a system of objects.

c. TRUE - You must know this!

d. TRUE - These are all non-conservative forces. You can add normal force to the list as well.

e. FALSE - Whether there is envy in a physicist's heart is not for us to tell; the evil found within one's heart is often vast and mysterious ... . We can however definitively say that a physicist classifies forces in order to analyze physical situations in accord with the classification. If only conservative-classified forces do work, then KE_i + PE_i = KE_f + PE_f. On the other hand if one or more non-conservative-classified forces are doing work, then KE_i + PE_i + W_nc = KE_f + PE_f.

f. FALSE - Not only must the force act upon the object, it must also be doing work upon the object. As you sit in your chair, there is a non-conservative force (normal force) acting upon your body. But since it does not do work (it's being assumed that you are not sitting in one of those fancy lounge chairs that has more controls than a TV set), your mechanical energy is not changing.

g. TRUE - This is a big principle. You must know this one!

h. TRUE - Non conservative forces would alter the total mechanical energy; that is, the PE + KE would not be a constant value.

i. TRUE (sort of) - This statement is true (sort of); when only conservative forces are doing work, an object has its kinetic energy transformed into potential energy (or vice versa) without the total amount of the two being altered. It would however be possible that work is not done by a non-conservative force and there be no transformation of energy at all; i.e., the object remains at rest. A conservative force must be doing work in order for there to be a transformation of energy.

j. TRUE - One would notice that the PE would begin to drop from 50 J to 0 J and that the KE would increase from 0 J to 50 J. And of course there would be a point at which the PE/KE would be distributed with 20 J to PE and 30 J to KE.

Answer: BGHK

a. FALSE - Kinetic energy depends upon the speed of the object; potential energy depends upon the position of the object.

b. TRUE - Kinetic energy depends upon speed. If there is no speed (the object is at rest), then there is no kinetic energy.

c. FALSE - If an object is on the ground, then it does not have potential energy (relative to the ground).

d. FALSE (sort of) - Kinetic energy depends upon mass and speed. Two objects of the same mass could have different weights if in a different gravitational field; so it is not appropriate to say that kinetic energy depends upon weight.

e. FALSE - Faster moving objects would have more kinetic energy than other objects of the same mass. However, another object could have less speed and make up for this lack of speed in terms of a greater mass.

f. FALSE - More massive objects would have more kinetic energy than other objects with the same speed. However, another object could have less mass and make up for this lack of mass in terms of a greater speed.

g. TRUE - Kinetic energy does not have a direction associated with it; it is a scalar quantity.

h. TRUE - Kinetic energy is directly related to the mass of an object.

i. FALSE - Kinetic energy is directly related to the square of the speed of an object. So a doubling of the speed would result in a quadrupling of the kinetic energy - the new KE would be 160 J.

j. FALSE - When it comes to kinetic energy, speed is doubly important (recall v²). So in this case, object A would have more kinetic energy. Doing the calculation yields 2 J for object A and 1 J for object B.

k. TRUE - Kinetic energy is determined by the equation 0.5•m•v². the quantity m is always positive. And even if v is negative, v² will always be positive. Therefore, kinetic energy can never be a negative value.

l. FALSE - If an object is falling at a constant velocity (i.e., the air resistance force equals the downward force of gravity), then there is not an increase in kinetic energy. It is true however that free-falling objects always increase their kinetic energy as they fall.

m. FALSE - The kinetic energy increases from 0 J to 2 J (0.5•1•2²); that's an increase by 2 J.

n. FALSE - Such an object will definitely gain or lose mechanical energy but not necessarily kinetic energy.

Answer: BDEFGH

a. FALSE - Potential energy has nothing to do with speed; an object could be moving at an elevated position. It is this elevation above zero level which gives an object potential energy.

b. TRUE - This is the definition of potential energy.

c. FALSE - Gravitational potential energy is dependent upon the mass of the object (PE_grav = m•g•h) but elastic potential energy is dependent upon the spring constant and the compression or stretch length of the spring (PE_elastic = 0.5•k•x²).

d. TRUE - The equation states that PE_grav = m•g•h; PE is dependent upon mass.

e. TRUE - The equation states that PE_grav = m•g•h; if the h is doubled, then the PE will be doubled as well.

f. TRUE - As objects free-fall, the height (h) decreases; subsequently, the PE decreases.

g. TRUE - The equation states that PE_grav = m•g•h; PE is directly related to height.

h. TRUE - The Joule (abbrev. J) is the standard metric unit of energy - all forms of energy.

i. FALSE - The final potential energy is calculated as PE = m•g•h = (1 kg)•(~10 m/s/s)•(1 m) = ~10 J.

j. FALSE - The final potential energy is calculated as PE = m•g•h = (1 kg)•(~10 m/s/s)•(6 m) = ~60 J; the loss in potential energy during this 4-m fall is -40 J.

k. FALSE - The object will either gain or lose mechanical energy, but not necessarily potential energy.

Answer: AEFGH

a. TRUE - This is the definition of mechanical energy.

b. FALSE - Heat or thermal energy is a non-mechanical form of energy. Potential and kinetic energy are the only forms of mechanical energy.

c. FALSE - The mechanical energy of an object is only conserved if non-conservative forces do not do work upon the object.

d. FALSE- If a non-conservative force does work upon an object, then the total mechanical energy of that object is changed. Energy will not be conserved.

e. TRUE - Tension does not do work upon the object and so the total mechanical energy is conserved. The presence of air resistance (a non-conservative force) does a little work and so one might notice a very slight change in mechanical energy.

f. TRUE - Friction is a non-conservative force and thus alters the total mechanical energy of an object.

g. TRUE - This is the conservation of energy principle and one that you need to firmly understand.

h. TRUE - If there is any change in the total mechanical energy of an object (whether a gain or a loss), then you know for certain that there is a non-conservative force doing work.

Answer: D < C < B < A

This is probably best done by performing a calculation of KE and comparing the results:

Object A: KE = 0.5•(5.0 kg)•(4.0 m/s)² = 40. J

Object B: KE = 0.5•(10.0 kg)•(2.0 m/s)² = 20. J

Object C: KE = 0.5•(1.0 kg)•(5.0 m/s)² = ~13 J (12.5 J)

Object D: KE = 0.5•(5.0 kg)•(2.0 m/s)² = 10. J

The order is evident once the calculations are performed.

Answer: C < A < D < B

This is probably best done by performing a calculation of PE and comparing the results.  Using the approximation that g = ~10 m/s/s gives much quicker results.

Object A: PE = (5.0 kg)•(~10 m/s²)•(2.0 m) = ~100 J

Object B: PE = (10.0 kg)•(~10 m/s²)•(3.00 m) = ~300 J

Object C: PE = (1.0 kg)•(~10 m/s²)•(5.0 m) = ~50 J

Object D: PE = (5.0 kg)•(~10 m/s²)•(4.0 m) = ~200 J

The order is evident once the calculations are performed.

Answer: C

The amount of work done by a force to displace an object is found from the equation

The force required to raise the car at constant speed is equivalent to the weight (m*g) of the car. Since the 2400-kg car weighs 2X as much as the 1200-kg car, it would require twice as much work to lift it the same distance.

Answer: A

For any given situation, the work done by a force can be calculated using the equation

where F is the force doing the work, d is the displacement of the object, and Theta is the angle between the force and the displacement. In this specific situation, the child is applying an upward force on the box (he is carrying it) and the displacement of the box is horizontal. The angle between the force (vertical) and the displacement (upward) vectors is 90 degrees. Since the cosine of 90-degrees is 0, the child does not do any work upon the box. A detailed discussion of a similar situation (the waiter and the tray of food) can be found at The Physics Classroom.

Answer: D

The use of the work-energy theorem and a simple analysis will yield the solution to this problem. Initially, there is only PE; finally, there is only KE. Assuming negligible air resistance, the kinetic energy of the diver upon hitting the water is equal to the potential energy of the diver on top of the board.

Substituting 500 N for m*g (500 N is the weight of the diver, not the mass) and 10 m for h will yield the answer of 5000 J.

Answer: A

During any given motion, if non-conservative forces do work upon the object, then the total mechanical energy will be changed. If non-conservative forces do negative work (i.e., F_nc*d*cos(Theta) is a negative number), then the final TME is less than the initial TME. In this case, air resistance does negative work to remove energy from the system. Thus, when the ball returns to its original height, their is less TME than immediately after it was thrown. At this same starting height, the PE is the same as before. The reduction in TME is made up for by the fact that the kinetic energy has been reduced; the final KE is less than the initial KE.

Answer: C

The use of the work-energy theorem and a simple analysis will yield the solution to this problem. Initially, there is only PE; finally, there is neither PE nor KE; non-conservative work has been done by an applied force upon the falling object. The work-energy equation can be written as follows.

Substituting 2500 N for m*g (2500 N is the weight of the driver, not the mass); 10.0 m for h; 0.100 m for the displacement of the falling object as caused by the upward applied force exerted by the post; and 90 degrees for Theta (the angle between the applied force and the displacement of the falling object) will yield the answer of 250000 N for F.

Answer: A

The KE of any object can be computed if the mass (m) and speed (v) are known. Simply use the equation

In this case, the 10-N object has a mass of approximately 1 kg (use F_grav = m*g). The speed is 1 m/s. Now plug and chug to yield KE of approximately 0.5 J.

Answer: B

Quite surprisingly to many, each ball would hit the ground with the same speed. In each case, the PE+KE of the balls immediately after being thrown is the same (they are thrown with the same speed from the same height). Upon hitting the ground, they must also have the same PE+KE. Since the PE is zero (on the ground) for each ball, it stands to reason that their KE is also the same. That's a little physics and a lot of logic - and try not to avoid the logic part by trying to memorize the answer.

Answer: D

An object at rest absolutely cannot have speed or velocity or acceleration. However, an object at rest could have energy if there is energy stored due to its position; for example, there could be gravitational or elastic potential energy.

Answer: C

Work involves a force acting upon an object to cause a displacement. The amount of work done is found by multiplying F*d*cos(Theta). The equation can be used for these two motions to find the work.

Answer: C

The power is the rate at which work is done (or energy is used). Power is found by dividing work by time. It requires the same amount of work to do these two jobs (see question #23) and the same amount of time. Thus, the power is the same for both tasks.

Answer: B

This is a relatively simple plug-and-chug into the equation P=W/t with W=1000. J and t=20.0 s.

Answer: ~150 J

The work done upon an object is found with the equation

In this case, the d=6.0 m; the F=24.5 N (it takes 24.5 N of force to lift a 2.5-kg object; that's the weight of the object), and the angle between F and d (Theta) is 0 degrees. Substituting these values into the above equation yields

Answer: 13.2 J

There are two methods of solving this problem. The first method involves using the equation

where F=20.8 N, d=0.636 m, and Theta=0 degrees. (The angle theta represents the angle betwee the force and the displacement vector;  since the force is applied parallel to the incline, the angle is zero.) Substituting and solving yields

The second method is to recognize that the work done in pulling the cart along the incline at constant speed changes the potential energy of the cart. The work done equals the potential energy change. Thus,

Answer: P = 849 Watts

Eddy's power is found by dividing the work which he does by the time in which he does it. The work done in elevating his 65.0-kg mass up the stairs is determined using the equation

where F = m*g = 637 N (the weight of the 65.0 kg object), d =1.60 m and Theta = 0 degrees (the angle between the upward force and the upward displacement). Solving for W yields 1019.2 Joules. Now divide the work by the time to determine the power:

Answers: See above table (answers in red)

The potential energy for every row can be found using the equation PE = m*g*h where m=510 kg and g = 9.8 m/s/s. In the first row, the total mechanical energy (KE + PE) equals 160 000 J (rounded). Since no work is done by non-conservative forces, the total mechanical energy must be 160 000 J in all the other rows. So the KE can be computed by subtracting the PE from 160 000 J. The velocity can be found using the equation:

where m=510 kg.  The algebraic rearrangement of this equation results in v = SQRT(2*KE/m).

Answers: (a) v = 101 m/s; (b) v = 77.2 m/s

a. Use the work energy theorem:

The PE_f can be dropped from the equation since the skier finishes on the ground at zero height. The KE_i can also be dropped since the skier starts from rest. The W_nc term is dropped since it is said that no work is done by non-conservative (external) forces. The equation simplifies to

The expressions for KE (0.5*m*v²) and PE (m*g*h) can be substituted into the equation:

where m=65.8 kg, h=521 m, g=9.8 m/s/s. Substituting and solving for v_f yields 101 m/s.

b. This equation can be solved in a similar manner, except that now the W_ext term is -140000 J. So the equation becomes

Now substituting and solving for v_f yields 77.2 m/s.

Description of Physical Situation

a. A force is applied to move a physics cart from the floor to the top of an inclined plane at a constant speed.

b. A physics student scurries up a flight of stairs at constant speed.

c. In a moment of unsupervised phun, a physics student hoists herself onto a staircase banister and accelerates down the banister. Ignore all friction forces.

d. A ball is dropped from rest from on the top of a hill and falls to the ground below. Ignore air resistance.

e. A ball leaves top of a hill with a large horizontal velocity. It falls to the ground below. Ignore air resistance.

f. A Hot Wheels car is at rest at an elevated position along an inclined plane; it is released and rolls to a position along the ground. Ignore air resistance.

g. A Hot Wheels car is in motion at the bottom of a hill when it hits a computer diskette box and skids to a stop.

h. A pendulum bob swings from its highest position to its lowest position.

i. A physics cart is released from rest at an elevated position along an inclined plane; it is released and rolls to a position along the incline approximately 5 cm from the bottom.

Answer: See table above

Note that whenever a non-conservative force is doing work (as in a, b, g), the total mechanical energy is changing.

However, when the only forces which do work are conservative forces, the total mechanical energy will remaining constant (i.e., be conserved). In such cases, energy will change form from potential to kinetic energy (or vice versa); yet the total amount of the two forms would not be changed.

Potential energy can be considered to change if an object changes its height; if the height decreases, then the potential energy decreases; it the height increases, then the potential energy increases.

Kinetic energy can be considered to change if an object changes its speed; if the speed decreases, then the kinetic energy decreases; it the speed increases, then the kinetic energy increases.

Answer: See above table; explanations given below.

There are a couple of major principles behind these bar charts: First, the initial energy (KE_i + PE_gi + PE_si) plus the work done by non-conservative forces (W_nc) must equal the final energy (KE_f + PE_gf + PE_sf); thus the heights of the bars on the left side of the chart must equal the height of the bars on the left side of the chart. Second, the bars must appropriately depict the form of energy and the relative amount for both the initial and final state. The above charts are not the only means of appropriately representing the motion. Other renditions which adhere to these two basic principles would be considered correct.

a. The car is initially at rest (no KE) and has some height above the floor (there is PE_g). The car is moving at the bottom of the track (there is KE) but it does not have any height (no PE_g). Since there is no friction nor air resistance, the W_nc term is zero.

b. Part b is similar to part a with the exception that friction and air resistance are to be considered in part b. So the analysis involves a consideration of the W_nc term. The W_nc term is negative since friction and air resistance are forces which oppose the motion of an object. The result is that energy has been removed from the system and the final kinetic energy bar is less than the initial potential energy bar.

c. If the tabletop (or floor) is considered the zero level, then the bob has gravitational potential energy in both its initial and final states. Yet the final PE_g is less than the initial PE_g since the final height (as shown) is less than the initial height. Since the air resistance is negligible, the W_nc term is zero. The difference in potential energy must be made up for in the form of kinetic energy. The bob would still be moving at the final position as shown.

d. This analysis is identical to part a except that the car has been replaced by a ball.

e. The dart is initially above the ground (the assumed zero-level) and finishes above the ground but a little lower than the initial height; thus, there is some PE_g to begin with and a little less PE_g in the final state. The springs are initially compressed, so there is some initial PE_s. The wall does work upon the dart to stop it; this would be negative work since the force is exerted in a direction opposite the motion of the dart. Initially the dart is not moving; so there is no initial KE. And after hitting the wall, it is not moving as well.

f. The worker is applying a force to displace the crate; this is a non-conservative force and so there will be a W_nc term. Since this applied force is in the direction of motion, the W_nc term will be positive. The crate is moving (it has KE) at a constant speed; since the speed is constant, the amount of KE is the same in the initial state as in the final state. Initially there is no gravitational potential energy (PE_g) since the crate starts on the ground; in the final state, there is some gravitational potential energy (PE_g) since the crate is at an elevated position.

Answer: 2.56 m/s

This is an example of energy transformation from potential energy at the highest point (the point of release) to kinetic energy at the lowest position. Since gravity is the only force doing work (tension acts perpendicular to the displacement so it does not do work), the total mechanical energy is conserved. So the energy conservation equation will be used.

Since the child starts from rest, the KE_i term can be canceled. And if we assign the lowest position as the zero-level, then the PE_f term can be canceled from the equation. The equation can be rewritten as

By substituting in expressions for kinetic and potential energy, the equation becomes

The mass cancels from each side of the equation, and algebra can be used to manipulate the equation to solve for v_f.

The crux of the problem becomes determining the initial height of the child. The length of the cables and the angle to which it is drawn back will be essential information in determining the initial height. The diagram at the right depicts the physical situation. The length of the cable is the hypotenuse of a triangle; the length of the side adjacent to the 17.4-degree angle can be determined using the cosine function. This adjacent side (a) is shorter than the length of the cable by an amount equal to the initial height (h_i). The work is shown below:

cos(17.4 deg) = a/L

a = L • cos(17.4 deg) = 6.95 m

h_i = L - a = 7.28 m - 6.95 mm = 0.333 m

Once h_i is found, it can be substituted into the above equation to solve for the final velocity:

v_f = SQRT ( 2 • 9.8 m/s/s • 0.33 m) = SQRT(6.53 m/s) = 2.56 m/s

Answer: 514 N

This is an example of work being done by a non-conservative force (the applied force of the mitt) upon a baseball in order to change its kinetic energy. So

The change in kinetic energy can be computed by subtracting the initial value (0.5 • m • v_i²) from the final value (0 J) .

The force can be determined by setting this value equal to the work and using the expression for work into the equation:

Answer: 116 m

During the entire descent down the hill, gravity is doing work on the skier and friction is doing negative work on the skier. Friction is a non-conservative force and will alter the total mechanical energy of the skier. The equation to be used is

If we designate the level area at the bottom of the slope as the zero level of potential energy, then PE_f is 0 J. Since the skier eventually stops (due to the effect of friction along the level area), the KE_f is 0 J. So the above equation becomes

The W_nc term has two parts; there is friction doing along the inclined plane and friction doing work along the level surface. Since these two sections of the motion have different normal forces and friction coefficients (and therefore friction forces), they will have to be treated separately. The graphic below depicts the free-body diagrams and the means by which the friction force can be determined.

By substituting values of mu and mass and g and theta into the above equations, one finds that the friction values are

These forces act upon the skier over different distances. In the case of the inclined plane, the distance (d) can be computed from the given incline angle and the initial height. The relationship is depicted in the diagram below. The sine function is used to relate the angle to the initial height and the distance along the incline. In the case of the level surface, the distance is the unknown quantity (x) which this problem calls for.

The distance d along the incline is

Now substitutions can be made into the work-energy equation and algebraic manipulation can be performed to solve for x: