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Impulse

In this experiment you will compare the impulse generated by a constant force applied over some time interval (t) with the change in the momentum of a glider on the air track. Impulse is defined as:
 

F D t = D (mv)

(1)

In the limit:
 

(2)

If the body is initially at rest and motion is in one-dimension:
 

(3)

Impulse is not often investigated experimentally even though it is involved in many important processes in our physical world.

If we have the case of a constant force it is easy to integrate the resulting impulse equation graphically. Consider the following:
Force (newtons) Time (sec)

Figure 1: Computation of impulse.

Using Figure 1, Impulse can be computed by measuring the area under a plot of F vs. t. Note that in this example you can use some simple geometric tools to find the area under the curve. Since you may not (in many cases) know the functional form of the integral, you can approximate (quite closely) by adding areas of simple triangles, rectangles or trapezoids. For the case above you might want to add a traingle for 0.0 to 0.4 seconds ( Area of triangle is 1/2 base X height) and height 8 Newtons ( area = 1/2 X 0.4 X 8.0 ~ 1.6 N-s ) and a trapezoid 0.4 sec to 0.5 sec, another trapezoid from 0.5 sec to about 0.8 sec, a little rectangle and another trapezoid. Simple geometry helps us do sophisticated numerical integration. In fact, using the trapezoidal rule we basically define what integrals do in the limit as the time interval approaches zero in the limit!

Integration is accomplished by counting the squares under the curve and adding the appropriate units. On the air track a force of this type (constant) can be achieved by passing a string from the glider over a pulley and down to a hanging weight.

Figure 2: A constant force can be achieved for the airtrack.

According to Newton's 2nd Law; for object m:
 

SFy = -may

(4)

so that:

T - mg = -may

(5)

Tension T is the force applied to the glider through the weight.

For the glider:
 

SFx = Max

(6)

so that:
 

T - f = Max

(7)

Assuming that friction is negligible (f 0) on the air track, we can solve for a:
 

a = mg/(M+m)

(8)

The displacement is a function of acceleration and time (where the initial velocity is zero):
 

d = 1/2 a t2

(9)

where t is the time interval. Thus the time interval is:
 

t = 2d/a = 2d(M+m)/mg

(10)

The force is merely Ma where M is the mass of the glider and a the acceleration of the falling weight and the glider.

OBJECTIVES:

1. To experimentally determine the impulse applied to a mass.
2. To experimentally determine the change in momentum of the mass.
3. To compare impulse applied to a body with its resulting change in momentum.

APPARATUS:

• 1 Air Track Apparatus
• 1 Pulley and String
• 1 Set of Masses
• 1 Glider
• 1 Photogate Timer

PROCEDURE:

1. Experimentally determine the mass of the glider.
2. Knowing this mass, choose a reasonable weight to add to the string to cause the glider to accelerate. This can be determined from consideration of equation (10); i.e. solve equation (10) for the unknown mass in terms of what distance and times you select.
3. The acceleration of the system is found from equation (8). Since the force is found from ma, the force is known. It is constant, independent of time and distance traveled. This is the force that will be applied to the glider.
4. Add mass M to the system. Make several trials of this. The glider should be initially at zero velocity, and accelerated when the mass M is added.
5. Measure the time interval while the glider travels a certain distance (across the photogate timer as used in Lab #2). Use this for several trials as a mean and t can be determined.
6. Plot the force (tension in string!) as time on linear graph paper. Be sure to label your axes properly.
7. Determine the total impulse transferred to the glider.
8. The total impulse is equal to the changes in momentum of the object. Compare this result with impulse calculated from the area under the curve in step 6.
9. Write up your lab manual neatly, including first this handout, then a description of special procedures, limitations, data, data analysis including graphs, summary, conclusions and recommendations in order.

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