PURPOSE

To demonstrate the validity of the work-energy theorem, and to use energy principles to measure the acceleration due to gravity, g.

EQUIPMENT  linear air track, DataLogger interface, glider, two photogates, shim, vernier calipers.

RELEVANT EQUATIONS

DISCUSSION

The work-energy theorem states that the total work W done on an object moving from point A to point B is equal to the difference in the kinetic energies of the object at A and B. The work done on the object is defined as:

where F_x is the magnitude of the component of the force parallel to the direction of motion of the object, and d is the distance the object is moved. The kinetic energy of an object is defined as:

where m is the mass of the object and v is its speed. In terms of these equations, the work-energy theorem can be written:

(1)

In this experiment, you will study this relationship on the air-track.
 

Figure 9-1 shows an object of mass m sitting on an inclined plane. If the plane is frictionless, the magnitude of the net force on the mass is equal to the component of the glider's weight acting parallel to the incline:

so the work done on m in moving it from A to B, a distance d along the incline, is:

W = Fnet d = (m g sin θ) d = m g Δy

(2)

where Δy = d sin θ is the vertical distance between points A and B. Because gravity is a conservative force the work done will always depend only on the vertical height between A and B, no matter how steep the incline.

Since the net work done by a conservative force depends only on the endpoints of the path taken, it is convenient to express its negative as a change in potential energy:

-Wcons = PEB - PEA = m g yB - m g yA = mg Δy

(3)

In the real world, usually there are other, non-conservative forces also doing work on the glider. We can take this into account using the work-energy theorem:

Substituting from Equation (3):

This result says that in general, if mechanical energy is not conserved, the sum of the changes in the kinetic and the potential energies equals the work done by the non-conservative forces, such as friction. Since in a case like ours friction does negative work (it is directed opposite to the displacement), the decrease in the potential energy on the way down will be somewhat larger than the increase in the kinetic energy. By the same reasoning, on the way up, the increase in the potential energy will be somewhat less than the decrease in the kinetic energy.

In terms of the glider set up with the glider going down, we can write:

where Δy = y_B - y_A = -h is negative in this case. In the case where the glider is going up the incline, v_B is less than v_A, so the change in kinetic energy is negative while Δy is positive.

If friction is absent, then mechanical energy is conserved, and the kinetic energy that the glider gains in traveling down the incline is exactly equal to the loss in potential energy as it moves lower. This trade off is reversed if the glider moves up the incline. In this experiment you will calculate these energy changes and use the result to estimate the force of kinetic friction.