6.3.  Gravitational Potential Energy

WORK DONE BY THE FORCE OF GRAVITY

The gravitational force is a well-known force that can do positive or negative work, and Figure 6.10 helps to show how the work can be determined. This drawing depicts a basketball of mass m moving vertically downward, the force of gravity mg being the only force acting on the ball. The initial height of the ball is h₀, and the final height is h_f, both distances measured from the earth’s surface. The displacement s is downward and has a magnitude of s=h₀–h_f. To calculate the work done on the ball by the force of gravity, we use with and since the force and displacement are in the same direction:

(6.4)

Equation 6.4 is valid for any path taken between the initial and final heights, and not just for the straight-down path shown in Figure 6.10. For example, the same expression can be derived for both paths shown in Figure 6.11. Thus, only the difference in vertical distances need be considered when calculating the work done by gravity. Since the difference in the vertical distances is the same for each path in the drawing, the work done by gravity is the same in each case. We are assuming here that the difference in heights is small compared to the radius of the earth, so that the magnitude g of the acceleration due to gravity is the same at every height. Moreover, for positions close to the earth’s surface, we can use the value of

Figure 6.10  Gravity exerts a force mg on the basketball. Work is done by the gravitational force as the basketball falls from a height of h0 to a height of hf.

Figure 6.11  An object can move along different paths in going from an initial height of h0 to a final height of hf. In each case, the work done by the gravitational force is the same since the change in vertical distance (h0–hf) is the same.

Since only the difference between h₀ and h_f appears in Equation 6.4, the vertical distances themselves need not be measured from the earth. For instance, they could be measured relative to a zero level that is one meter above the ground, and would still have the same value. Example 7 illustrates how the work done by gravity is used in conjunction with the work-energy theorem.

A gymnast springs vertically upward from a trampoline as in Figure 6.12a. The gymnast leaves the trampoline at a height of 1.20 m and reaches a maximum height of 4.80 m before falling back down. All heights are measured with respect to the ground. Ignoring air resistance, determine the initial speed v0 with which the gymnast leaves the trampoline. Figure 6.12  (a) A gymnast bounces on a trampoline. (©James D. Wilson Liaison/Getty Images) (b) The gymnast moves upward with an initial speed v 0 and reaches maximum height with a final speed of zero. Reasoning  We can find the initial speed of the gymnast by using the work-energy theorem, provided the work done by the net external force can be determined. Since only the gravitational force acts on the gymnast in the air, it is the net force, and we can evaluate the work by using the relation Solution Figure 6.12b shows the gymnast moving upward. The initial and final heights are and respectively. The initial speed is v0 and the final speed is since the gymnast comes to a momentary halt at the highest point. Since the final kinetic energy is and the work-energy theorem becomes The work W is that due to gravity, so this theorem reduces to Solving for v0 gives

Gravitational Potential Energy

We have seen that an object in motion has kinetic energy. There are also other types of energy. For example, an object may possess energy by virtue of its position relative to the earth; such an object is said to have gravitational potential energy. A pile driver, for instance, is used by construction workers to pound “piles,” or structural support beams, into the ground. The pile driver contains a massive hammer that is raised to a height h and then dropped (see Figure 6.13). As a result, the hammer has the potential to do the work of driving the pile into the ground. The greater the height of the hammer, the greater is the potential for doing work, and the greater is the gravitational potential energy.

Figure 6.13  In a pile driver, the gravitational potential energy of the hammer relative to the ground is .

Now, let’s obtain an expression for the gravitational potential energy. Our starting point is Equation 6.4 for the work done by the gravitational force as an object moves from an initial height h₀ to a final height h_f:

(6.4)

This equation indicates that the work done by the gravitational force is equal to the difference between the initial and final values of the quantity mgh. The value of mgh is larger when the height is larger and smaller when the height is smaller. We are led, then, to identify the quantity mgh as the gravitational potential energy. The concept of potential energy is associated only with a type of force known as a “conservative” force, as we will discuss in Section 6.4.

DEFINITION OF GRAVITATIONAL POTENTIAL ENERGY

The gravitational potential energy PE is the energy that an object of mass m has by virtue of its position relative to the surface of the earth. That position is measured by the height h of the object relative to an arbitrary zero level:  (6.5)  SI Unit of Gravitational Potential Energy: joule (J)

Gravitational potential energy, like work and kinetic energy, is a scalar quantity and has the same SI unit as they do, the joule. It is the difference between two potential energies that is related by Equation 6.4 to the work done by the force of gravity. Therefore, the zero level for the heights can be taken anywhere, as long as both h₀ and h_f are measured relative to the same zero level. The gravitational potential energy depends on both the object and the earth (m and g, respectively), as well as the height h. Therefore, the gravitational potential energy belongs to the object and the earth as a system, although one often speaks of the object alone as possessing the gravitational potential energy.

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