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.