1D-Motion: Measuring g©98
Experiment 2
Objective: To describe the motion of a freely-falling object using its position, velocity and acceleration, then comparing the acceleration to the expected value.
Discussion:
In this experiment we will learn how to describe the motion of an object. In particular, we will observe, record, and study the motion of an object falling in the gravitational field of earth. The object, whose motion we wish to study, is released from a position of rest and is allowed to fall freely for a distance of several feet. During the fall, its position is recorded at successive equal intervals of time, by means of a spark record on a paraffin wax-coated paper tape. From this position and time record, a complete analysis of the motion can be made. Figure 1 shows a schematic of the Brehm apparatus:
Refer to Figure 1 during the following discussion for the procedure used to obtain the position-indicating spark record on the tape. Two carefully spaced vertical wires are mounted so that the plummet falls freely between them. Connected to the two wires is a periodic high-voltage source which produces a spark of very short duration and at a repetition rate of 60 sparks per second. This rate is easily found in the same 60 Hz frequency that AC power is delivered from the outlet.
The voltage used is not sufficient for the spark to jump the full distance between the wires. However, a sharp projecting metal shoulder on the plummet provides a shortened air path for the spark to jump and ensures that the spark will always jump at the same position on the plummet. In its jump from the shoulder on the plummet to the wire, the spark passes through a paraffin wax-coated paper tape stretched tightly between the wire and the path of the plummet. Each spark punctures the specially prepared tape, leaving a small hole, easily identified by the area of melted paraffin around the spark hole.
The trace thus obtained consists of a series of marks of increasing separation, each mark giving the position of the plummet at that time. Inasmuch as no attempt is made in this experiment to release the object at the precise instant of a spark, the initial time of release is unknown. Thus the analysis is made on the basis of the time and distance intervals from one of the first clear marks. Figure 2 reveals a portion of a typical tape record of the motion of the falling plummet.
Once the tape is made, we have a complete record of the position of the plummet at particular times. The question is then how to use the information to describe the motion. The obvious first thing to do is to make a plot of the position s as a function of the time t. If asked to describe the motion of the falling plummet, we could just submit this plot. But suppose we are then quizzed about the speed of the object. We know that the speed is related to the position and time by
(1)
This says, if we know the change in position Ds of the object for a certain change in time Dt then we can find the speed of that time over the time interval Dt. The information Ds and Dt is readily obtained from the data taken off the tape, hence v is readily calculated in a chart. But suppose we are then quizzed about the acceleration of the object. We know that the acceleration is related to the speed and time by
(2)
This says, if we know the change in speed Dv of the object for a certain change in time Dt then we can find the acceleration. The acceleration is, of course, uniform, which is the case for an object falling in the gravitational field of the earth. We can now use this information on s and Dt, to calculate Ds, v, Dv and a at various times, and compare the value of a to the expected value of g.
EXERCISES:
1. Your instructor will show you the details of the apparatus and help you obtain your own record of motion for the freely-falling body.
2. Lay the tape flat on the laboratory table, taping at the ends with the masking tape, and examine it for completeness. There should be 25-30 dots along the length of the tape. This means that we have available as data the position of the plummet at 30 or so different times. We don’t have time to work with all 30 dots, so lets agree to work with every third dot beginning with the first or second small dot after the initial smudge. Keep in mind that the time interval between each successive data dot is 1/60 sec making the time interval between every third dot is 1/20 (or 0.05) sec.
3. Circle the beginning dot and label it s0 =0, and t =0. Then go down the tape, labeling every third dot. Mark each one s1, s2, s3,…
4. Take a very sharp pencil and using the ruler, draw a line through the center of the dot that is perpendicular to the path of the track. Now use the ruler to measure in cm the distance from the first dot to the first few chosen dots, and then after 30cm, use the meter stick to determine the rest. Refer to Figure 2 for clarification.
5. Fill in your data and calculation sheet and determine the value of the acceleration due to gravity. Give the % error between your value and 980 cm/s².
6. Now make a plot of s as a function of time t. When the graph is complete, confirm for yourself that the graph is a parabola of the form s = 0.5gt².
7. Make two additional plots, one each of v and a as a function of t. What do these plots tell you about the velocity and the acceleration of the falling body?
Data Sheet - Experiment 2
sn / Cumulative
Elapsed Time
t
(s) / Total Distance
sn-s0
(cm) / Distance between points Ds=sn-sn-1
(cm) / Speed
v=Ds/Dt
(cm/s) / Difference in successive speeds
Dv=vn-vn-1
(cm/s) / Acceleration a=Dv/Dt
(cm/s2)
s0 / 0.00 / 0.00
s1 / 1/20
s2 / 2/20
s3 / 3/20
s4 / 4/20
s5 / 5/20
s6 / 6/20
s7 / 7/20
s8 / 8/20
s9 / 9/20
s10 / 10/20
s11 / 11/20
Average acceleration: (m/s2)
% Error between average a and g:
Data Sheet - Experiment 2 cont.
Data Sheet - Experiment 2 cont.
Data Sheet - Experiment 2 cont.
2-8