Lesson 2 : The Work - Energy Theorem

Explanation
The ball is losing height (falling) and gaining speed. Thus, the internal or conservative force (gravity) transforms the energy from PE (height) to KE (speed).

Explanation
The skier is losing height (the final location is lower than the starting location) and gaining speed (the skier is faster at B than at A). Thus, the internal force or conservative (gravity) transforms the energy from PE (height) to KE (speed).

Explanation
The ball is gaining height (rising) and losing speed (slowing down). Thus, the internal or conservative force (gravity) transforms the energy from KE (speed) to PE (height).

Explanation
The jumper is losing speed (slowing down) and the bunjee cord is stretching. Thus, the internal or conservative force (spring) transforms the energy from KE (speed) to PE (a stretched "spring"). One might also argue that the gravitational PE is decreasing due to the loss of height.

Explanation
The spring changes from a compressed state to a relaxed state and the dart starts moving. Thus, the internal or conservative force (spring) transforms the energy from PE (a compressed spring) to KE (speed).

Explanation
Initially:

PE = 0 J (the car's height is zero)
KE = 0.5*1000*(25)^2 = 312 500 J

Finally:

PE = 0 J (the car's height is zero)
KE = 0 J (the car's speed is zero)



The work done is (8000 N) • (d) • cos 180 = - 8000*d

Using the equation,

TMEi + Wext = TMEf
312 500 J + (-8000 • d) = 0 J

Using some algebra it can be shown that d=39.1 m

Explanation
Initially:

PE = 0 J (the car's height is zero)
KE = 0.5*6000*(20)^2 = 1 200 000 J



Finally:

PE = 0 J (the car's height is zero)
KE = 0.5*6000*(5)^2 = 75 000 J



The work done is F • 20 • cos 180 = -20•F

Using the equation,

TMEi + Wext = TMEf
1 200 000 J + (-20*F) = 75 000 J

Using some algebra, it can be shown that 20*F = 1 125 000 and so F = 56 250 N

Explanation
The question pertains to the can of peaches; so focus on the can (not the cart).

Initially:

PE = 0.25 kg * 9.8 m/s/s * 2 m = 4.9 J
KE = 0 J (the peach can is at rest)



Finally:

PE = 0 J (the can's height is zero)
KE = 0 J (the peach can is at rest)



The work done is 500 N*d*cos 180 = -500*d

Using the equation,

TMEi + Wext = TMEf
4.9 J + (-500*d) = 0 J

Using some algebra, it can be shown that d = 0.0098 m (9.8 mm)

Explanation
The only force doing work is gravity. Since it is an internal or conservative force, the total mechanical energy is conserved. Thus, the 100 J of original mechanical energy is present at each position. So the KE for A is 50 J.

The PE at the same stairstep is 50 J (C) and thus the KE is also 50 J (D).

The PE at zero height is 0 J (F and I). And so the kinetic energy at the bottom of the hill is 100 J (G and J).

Using the equation KE = 0.5*m*v2, the velocity can be determined to be 7.07 m/s for B and E and 10 m/s for H and K.

The answers given here for the speed values are presuming that all the kinetic energy of the ball is in the form of translational kinetic energy. In actuality, some of the kinetic energy would be in the form of rotational kinetic energy.

Explanation
When work is done on an object by an external or nonconservative force, there will be a change in the total mechanical energy of the object. This is because work is the transfer of energy, and when work is done on an object, it can either increase or decrease the object's mechanical energy. The total mechanical energy of an object is the sum of its kinetic energy and potential energy, so any change in work will lead to a corresponding change in the object's total mechanical energy.

Explanation
The explanation for the given correct answer is that according to the principle of conservation of energy, whenever work is done on an object by an external or nonconservative force, there will be a change in the total mechanical energy of the object. This means that the mechanical energy is not conserved in such cases. However, if only internal forces are doing work and there is no work done by external forces, then the total mechanical energy remains constant and is said to be conserved. Therefore, the statement that there is a relationship between work and mechanical energy change is true.

Explanation
The equation provided is a statement of the conservation of mechanical energy. It states that the initial amount of total mechanical energy, which includes both kinetic and potential energy, plus the work done by external forces, is equal to the final amount of total mechanical energy. This means that in a closed system, the total mechanical energy remains constant. Therefore, the given statement is true.

Explanation
The statement suggests that work-energy problems should not be treated as mere mathematical problems. This implies that there is more to these problems than just solving equations and calculations. Work-energy problems involve understanding and applying the principles of work and energy, which are fundamental concepts in physics. By treating these problems as mere mathematical problems, one may overlook the physical significance and implications of the concepts involved. Therefore, the correct answer is False.

Explanation
The answer is D. The total mechanical energy (i.e., the sum of the kinetic and potential energies) is everywhere the same whenever there are no external or nonconservative forces (such as friction or air resistance) doing work.

Explanation
The answer is C. Since the total mechanical energy is conserved, kinetic energy (and thus, speed) will be greatest when the potential energy is smallest. Point B is the only point that is lower than point C. The reasoning would follow that point B is the point with the smallest PE, the greatest KE, and the greatest speed. Therefore, the object will have less kinetic energy at point C than at point B (only).

Explanation
The answer is B. Gravitational potential energy depends upon height (PE=m*g*h). The PE is a minimum when the height is a minimum. Position B is the lowest position in the diagram.

Explanation
The statement is true because analyzing the initial and final states of an object allows us to determine the different forms of energy present. By examining the object's initial state, we can identify the initial forms of energy such as kinetic or potential energy. Similarly, by analyzing the object's final state, we can determine the final forms of energy. This analysis helps in constructing work-energy bar charts accurately, as it provides information about the energy transformations and transfers that occur during the process.

Explanation
In constructing work-energy bar charts, analyzing the forces acting upon the object during motion is necessary to determine if external forces are doing work. This analysis helps to determine whether the work done by these external forces is positive or negative. By understanding the forces and their effects on the object, one can accurately represent the energy transfers and transformations in the bar chart. Therefore, the statement is true.

Explanation
In constructing work-energy bar charts, the exact height of the individual bars is not important because what matters is the comparison between the heights on the left and right sides of the chart. The purpose of the chart is to illustrate the presence and absence of various forms of energy in the initial and final states of the object. The sum of the heights on the left side of the chart represents the initial state, while the sum of the heights on the right side represents the final state. The balancing of the heights on both sides ensures that energy is conserved in the system. Therefore, the statement is true.

Explanation
When only internal forces are doing work and no work is done by external forces, the total mechanical energy of an object remains constant. This means that there is no change in the sum of the object's kinetic and potential energies. Therefore, the total mechanical energy is said to be conserved. On the other hand, external forces have the ability to change the total mechanical energy of an object, making them nonconservative forces. Internal forces, however, do not change the total mechanical energy of an object and are therefore referred to as conservative forces.