Lesson 2: The Work-Energy Theorem

Internal vs. External Forces

There are a variety of ways to categorize all the types of forces. In a previous unit, it was mentioned that all the types of forces can be categorized as contact forces or as action-at-a-distance forces. Whether a force was categorized as an action-at-a-distance force was dependent upon whether or not that type of force could exist even when the objects were not physically touching. The force of gravity, electrical forces, and magnetic forces were classic examples of forces which could exist between two objects even when they are not physically touching. In this lesson, we will learn how to categorize forces based upon whether or not their presence is capable of changing an object's total mechanical energy. We will learn that there are certain types of forces, which when present and when involved in doing work on objects will change the total mechanical energy of the object. And there are other types of forces which can never change the total mechanical energy of an object, but rather can only transform the energy of an object from potential energy to kinetic energy (or vice versa). The two categories of forces are internal versus external forces.

Forces can be categorized as internal forces or external forces. There are many sophisticated and worthy ways of explaining and distinguishing between internal and external forces. Many of these ways are commonly discussed at great length in physics textbooks. For our purposes, we will merely say that external forces include applied forces, normal forces, tensional forces, friction forces, and air resistance forces. For our purposes, internal forces include gravitational forces, magnetic forces, electrical forces, and spring forces.

Internal Forces	External Forces
Fgrav Fspring	Fapp Ffrict Fair Ftens Fnorm

The significance of categorizing a force as internal or external is related to the ability of that type of force to change an object's total mechanical energy when it does work upon an object. When work is done upon an object by an external force, the total mechanical energy (KE + PE) of that object is changed. If the work is "positive work", then the object will gain energy. If the work is "negative work", then the object will lose energy. The gain or loss in energy can be in the form of potential energy, kinetic energy, or both. Under such circumstances, the work which is done will be equal to the change in mechanical energy of the object; this principle will be discussed in great detail later in this lesson.

When work is done upon an object by an internal force (for example, gravitational and spring forces), the total mechanical energy (KE + PE) of that object remains constant. In such cases, the object's energy changes form. For example, as an object is "forced" from a high elevation to a lower elevation by gravity, some of the potential energy of that object is transformed into kinetic energy. Yet, the sum of the kinetic and potential energies remain constant. This is referred to as energy conservation and will be discussed in detail later in this lesson. When the only forces doing work are internal forces, energy changes forms - from kinetic to potential (or vice versa); yet the total amount of mechanical is conserved.

In the following descriptions, the only forces doing work upon the objects are internal forces - gravitational and spring forces. Thus, energy is transformed from KE to PE (or vice versa); however, the total mechanical energy is conserved. Read each description and indicate whether energy is transformed from KE to PE or from PE to KE. Depress the mouse on the "pop-up menu" to check your answers.

	Description of Motion	KE to PE or PE to KE? Explain.
1.	A ball falls from a height of 2 meters in the absence of air resistance.	Depress mouse to view answer. PE to KE The ball is losing height (falling) and gaining speed. Thus, the internal force (gravity) transforms the energy from PE (height) to KE (speed).
2.	A skier glides from location A to location B across the friction free ice.	Depress mouse to view answer. PE to KE The skier is losing height (the final loc'n is lower than the starting loc'n) and gaining speed (the skier is faster at B than at A). Thus, the internal force (gravity) transforms the energy from PE (height) to KE (speed).
3.	A baseball is traveling upward towards a man in the bleachers.	Depress mouse to view answer. KE to PE The ball is gaining height (rising)) and losing speed (slowing down). Thus, the internal force (gravity) transforms the energy from KE (speed) to PE (height).
4.	A bungee chord begins to exert an upward force upon a falling bungee jumper.	Depress mouse to view answer. KE to PE The jumper is losing speed (slowing down) and the bunjee chord is stretching. Thus, the internal force (spring) transforms the energy from KE (speed) to PE (a stretched "spring"). One might also argue that the grav. PE is decreasing due to the loss of height.
5.	The spring of a dart gun exerts a force on a dart as it is launched from an initial rest position.	Depress mouse to view answer. PE to KE The spring changes from a compressed state to a relaxed state and the dart starts moving. Thus, the internal force (spring) transforms the energy from PE (a compressed spring) to KE (speed).

NOTE: Perhaps at this time you might find it useful to review the lessons on kinetic energy and potential energy.

When work is done by external forces, the total mechanical energy of the object is altered. The work that is done can be "+ work" or "- work" depending on whether the force doing the work is directed opposite the object's displacement or in the same direction as the object's displacement. If the force and the displacement are in the same direction, then "+ work" is done on the object; the object subsequently gains mechanical energy. If the force and the displacement are in the opposite direction, then "- work" is done on the object; the object subsequently loses mechanical energy.

The following descriptions involve external forces (frictional, applied, normal, and tensional forces) acting upon an object. Read the description and indicate whether the object gained energy ("+ work") or lost energy ("- work"). (NOTE: If this is part is difficult, review the section on work.) Then, indicate whether the gain or loss of energy resulted in a change in the object's kinetic energy, potential energy, or both. Depress the mouse on the "pop-up menu" to view answers.

Description	+ or - Work?	Change PE or KE or Both?
Megan drops the ball and hits an awesome forehand. The racket is moving horizontally as the strings apply a horizontal force while in contact with the ball.	View answer. The work is +. The force is to the right and the displacement is to the right. F and d are in the same direction. Thus, positive work is done.	View answer. Kinetic energy. The applied force of the racket causes the ball to gain speed. Thus, the external force alters the kinetic energy of the ball.
A baseball player hits the ball into the outfield bleachers. During the contact time between ball and bat, the bat is moving at a 10 degree angle to the horizontal.	View answer. The work is +. The force is up a nd to the right and the displacement is up and to the right. F and d are in the same direction. Thus, positive work is done.	View answer. Both The applied force of the bat causes the ball to gain both height and speed. Since the force has an up component, it contributesto a height change. Thus, the external force alters the both the kinetic energy and the potential energy of the ball.
Rusty Nales pounds a nail into a block of wood. The hammer head is moving horizontally when it applies force to the nail.	View answer. The work is +. The force is to the left and the displacement is to the left. F and d are in the same direction. Thus, positive work is done.	View answer. Kinetic energy. The applied force of the hammer causes the nail to gain speed. Thus, the external force alters the kinetic energy of the nail.
The frictional force between highway and tires pushes backwards on the tires of a skidding car.	View answer. The work is -. The force is to the left and the displacement is to the right. F and d are in the opposite direction. Thus, negative work is done.	View answer. Kinetic energy. The friction force on the car causes the car to lose speed. Thus, the external force alters the kinetic energy of the car.
A diver experiences a horizontal reaction force exerted by the blocks upon her feet at start of the race.	View answer. The work is +. The force is to the right and the displacement is to the right. F and d are in the same direction. Thus, positive work is done.	View answer. Kinetic energy. The applied force of the starting blocks causes the diver to gain speed. Thus, the external force alters the kinetic energy of the diver. (NOTE: there is another force - gravity - which changes the diver's height; but the blocks are not responsible for this height change.)
A weightlifter applies a force to lift a barbell above his head at constant speed.	View answer. The work is +. The force is up and the displacement is up. F and d are in the same direction. Thus, positive work is done.	View answer. Potential energy. The applied force of the causes the barbell to gain height. Thus, the external force alters the potential energy of the barbell.

Note that in the five situations described above, a horizontal force can never change the potential energy of an object. Horizontal forces cannot cause vertical displacements. The only means by which an external force can contribute to a potential energy change is if the force has a vertical component. Potential energy changes are the result of height changes and only a force with a vertical component can cause a height change.

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