Recall that work is calculated with the formula W = Fs cos θ where θ is the angle between the direction of the force and the direction of the object's motion.

 

The work energy theorem states that the work done on an object equals the change in its kinetic energy, W = ΔKE.

 

When we examine an object dropped from rest in a state of freefall we notice that as gravity accelerates the object towards the surface of the earth, not only does the object's kinetic energy increases as its speed increases but its potential energy decreases as its height above the surface of the earth decreases. This is a general property of any conservative force: work done results in a subsequent decrease in the object's potential energy.

 

 

 

Let's look at another example to highlight the inclusion of this "negative sign." Since we are working near the surface of the earth and our objects will not go beneath the surface, we are going to use the floor (the position A in the above diagram) as our zero position. Our initial discussions will employ the equation PE_g = mgh where h = r - R_E. That is, our h will represent the difference between an object's initial distance from the center of the earth and the earth's surface. Later in the lesson we will see that this is a reasonable equation to calculate an object's gravitational potential energy when r << R_E.

 

Note that the changes in an object's potential energy only depend on comparing its starting position and its ending position, not on whether it does or does not pass through various points in-between - that is, gravity is a path-independent force. The block's final change in potential energy is the same whether it follows the path with the intermediate stops B, C and D or whether it is directly taken from E to A.  The height of the post is the same.

 

When lowering an object [(y_f - y_o ) < 0], θ equals 0º.

 

Work_{by gravity} = mg(-h) cos 0º
                    = mg(-h)(+1)
                    = - mgh

 

 

 

 

For the remainder of the lesson we will use the variable U to represent potential energy as we derive a generalize equation for an object's potential energy at and arbitrary R > R_E. Let's assume that our point mass begins at an extremely distant location infinitely far away and is being gravitationally attracted to a position closer to the earth's surface, R.

 

The vector nature of the gravitational force, F_G, is written as

 

 

where  is a unit vector pointing from M₁ to M₂. Since gravity pulls M₂ towards M₁, we use the negative to denote that the direction of the force is the reverse of .

 

Recall from our discussion at the top of this page that the work done by gravity equals the negative change in the object's gravitational potential energy.

 

 

Since our force is not constant, we will have to use integration to calculate the work that will need to be done.

 

 

Putting together our two expressions for work we have

 

 

Simplifying the left side of our equation we get

 

 

Since our functional value depends on the ratio (1/R) it is natural to define the potential energy (U_o) as being 0 at infinity since the limit as R approaches infinity of (1/R) equals 0.

 

We now have our final expression for the potential energy of a satellite at any radius R from the center of a planet.

 

 

A graph of U(R) vs R is shown below.

 

 

As expected, all values of the potential energy are negative, approaching the value of zero as R approaches infinity. Because of this we say that the mass is trapped in an "energy well" - that is, it will have to be given additional energy from an external source if it is to escape gravity's attraction.

 

 

Notice that the simplified expression for an object's potential energy near the surface of the earth that we used in the first part of our lesson, PE = mgh, can be derived from our generalized formula.

 

Recall that the gravitational field strength is given by

 

 

The potential energy at the surface of the earth is

 

 

Due to the almost linear nature of our graph near the planet's surface we can approximate the potential energy for small heights (h<<R_E) above the earth's surface as

 

 

The change in potential energy for a falling mass would be

 

 

 

 

When a projectile is thrown upwards off the surface of the earth, the decrease in its kinetic energy equals the increase in its potential energy. At any given time, its total energy will equal

 

total energy = K + U

 

If we wanted an unpowered projectile to escape from the earth's surface, it would need to have a total energy equal to or greater than zero to get out of the earth's gravity well.