PURPOSE

In this experiment, you will be dealing with two different types of simulation: the inclined plane, and the suspended block/horizontal block system (half-Atwood Machine) .

RELEVANT EQUATIONS

REFERENCES Cutnell & Johnson: pp. 154-174

DISCUSSION

A: Inclined Plane

We first examine the energy balance involved when a block slides down an incline, as shown in Figure 8-1.

We assume that the block passes through two markers as it slides down the incline, and that the markers are separated by a distance d as the ramp falls through a distance h. You can verify that the angle of incline is related to these parameters through the relation:

The physical law we will investigate is often referred to as the Work-Energy theorem. Simply stated, it relates the change in the kinetic energy as the block moves from point A to point B, DK = K_B - K_A, to the net work done by all external agents. Since the work done is equal to the net force component in the direction of travel times the distance covered, it is expressed as:

where we have chosen the x-axis to lie along the incline. In general, there are two forces with components in the x-direction, the parallel component of the block's weight mg sinq and kinetic friction f.
 
 

We assume that friction can be neglected, since the actual incline that will be used in the practical part of this experiment is an air track. The work done by the net force as the block moves down the incline through a distance d is therefore: W = mgd sin q= mgh. The change in kinetic energy is:

Equating the two leads to the following relation:
 
 

                                                                                                    (1)

 The first equation says that the gain in kinetic energy is equal to the loss in gravitational potential energy. The square of the velocity of the block is therefore directly proportional to the vertical distance h through which it falls. The proportionality constant is twice the acceleration of gravity, 2g. The objective of the simulation will be to measure the velocity of the sliding block at two points and relate the difference in the square of the velocities to the vertical height through which the block falls.

B: Half Atwood's Machine (Linked Horizontal and Falling Masses)

In this section you will be working with the same apparatus that was used in the experiment on Newton's Second Law. The simulation diagram is shown below in Figure 8-2.

Assuming no kinetic friction, work is done on the horizontal block only by the tension T in the string. On the plummet, there are two forces that do work: the tension T and the weight of the block m₂g. If the system moves through a distance d, then the total work done is:

Using the Work-Energy theorem, we can equate the work to the change in the kinetic energy as follows:

This can be simplified to give:

                                                                                         (2)

 In this case, the square of the speed (of either block) is directly proportional to the distance through which they travel. The coefficient is twice the acceleration of gravity g, multiplied by the ratio of the falling mass to the total mass.