April 01, 2015 Centripetal Force with a Motor Lab

Objective: Find the relationship between the angle and the angular velocity.

The Set-up: 

The apparatus is a motor mounted on a surveying tripod. A long bar is connected vertically to the motor.  And a meter stick is mounted on the long bar. At the end of the meter stick is a string that has a weight attached. 

Data Collection:

Here are a few things that we measure:

The length of the string (L) =165.4cm ± 1 cm

The radius of the circle made by string/mass (R) =97.5 cm ± 1 cm

The height of the horizontal rod (H) =200cm ± 1 cm

The angle and height (h) are the variables that can be obtained through experiment. After collecting the data, we can use them to find angular velocity.

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Next,  we come up with a model of the angle and the angular velocity.

After finding the model of the angle and the angular velocity, we begin our data collection. We find angular velocity by divide a number of rounds that the weight made with the time. As for the angle, we obtain by recording the height that the weight reaches. Then we can calculate the angle using inverse cosine. 

Here is our data.

Table 1

Trial	Height of the mass, h (cm)	Period, T (sec)
1	47.3± 0.5	3.70
2	62.4 ± 0.5	3.26
3	85.0 ± 0.5	2.84
4	118.5 ± 0.5	2.20
5	140.8 ± 0.5	1.93
6	161± 1	1.56

Sample Calculation

Table 2

Trial	Angle	Theoretical Angular Velocity (rad/sec)	Experimental Angular Velocity (rad/sec)
1	22.6	1.59	1.70
2	33.7	1.84	1.93
3	46.0	2.17	2.21
4	60.5	2.68	2.86
5	69.0	3.19	3.26
6	76.4	3.96	4.03

The sample calculation shows the theoretical angular velocity. Theoretical angular velocity is the angular velocity that the mass should have under ideal situation.

Since the apparatus is not under ideal situation, there is the experimental angular velocity. The experimental angular velocity can be found by:

Angular velocity = 2pi/T.

T is period, it is the time it takes for one revolution. T is recorded in table 1.

For example, the experimental angular velocity of trial 1 would be 2pi/3.70 = 1.697≈1.70 rad/sec.

Finally, we input both the theoretical and experimental angular velocity into loggerpro, we get a graph that looks like the following. 

Experimental angular velocity vs theoretical angular velocity graph

Here we see that our experimental value, the red line, did not quite match the theoretical value, the black line. The reason is because there are some uncertainties involved in this apparatus. First of all, when we measure the length of the string, the radius and the height of the horizontal rod, there is an uncertainty value of ±1cm since we measure with the meter stick. Secondly when measuring the height of the mass (h), there is an uncertainty value of ±0.5 cm. When we measure the period, the time it takes for one revolution, we count the revolution with our eyes. There is some uncertainty involves since our counting are not exact. Also, there are some system error involves. When the apparatus turns, the horizontal rod, which is a meter stick, does not turn in a horizontal manner, sometimes it can wobble a little. Also, there is air resistance. The air resistance can slow down the angular velocity.

Summary

We begin this experiment with an apparatus that involves the rotation of the mass hanging on a string. Our goal is to find the relationship between the angle and the angular velocity. Firstly, we measure a few things that are always constant; they are the length of the string, the radius and the height of the horizontal rod. Then we find a formula for the angle and a formula for the angular velocity. To find the angle, we need the height of the mass. We get the height of the mass through experiment; it is recorded in table 1. After finding the angle, we can calculate the theoretical angular velocity using the formula we derived. We also take down the period, or the time that the mass takes to make one revolution. The period is for finding the experimental angular velocity. To find the experimental angular velocity, we simply divide 2pi by the period. The angle, theoretical angular velocity and experimental angular velocity are recorded in table 2. Then, we compare the theoretical value with the experimental value. We make a graph showing the difference. There is a small discrepancy between the two, it is due to the uncertainties involve in this lab.