# Simple Pendulum

Topics: Pendulum, Simple harmonic motion, Kilogram Pages: 15 (1507 words) Published: December 12, 2012
The Simple Pendulum
According to theory1, the period of a simple pendulum is independent of mass and is dependent on the square root of its length; i.e. T = 2B%L/g, where T is the period of the pendulum, L is the length of the pendulum and g is the acceleration of gravity. A simple pendulum is set up in the laboratory to test this assertion. A vertical shaft is placed in a lab table using the provided receptacle. A horizontal bar is then connected to the vertical shaft with a clamp, and a string is attached to the horizontal bar. A mass is attached to the other end of the string and pulled back and let go, so that it executes (approximately) simple harmonic motion. The time required to complete 30 cycles (t) is measured with a stopwatch and recorded. To improve accuracy, three trials are completed for each measurement. The average of the recorded values of t for the three trials is then divided by 30 to obtain the period (T) of the motion.

In the first part of the experiment, the length of the string is kept constant and different masses are attached to its end to determine the dependence (or lack thereof) of the period on mass. In the second part, the length of the string is varied, while keeping the mass constant in order to determine the dependence of the period on length. Graphs are drawn for period vs. mass and for period vs. length. A graph of log T vs. Log L is then drawn in order to determine the specific power dependence of T on L. The value obtained for this power is compared to the expected value of 1/2.

Finally, a graph of T vs. %L is drawn. According to the equation above, the slope of this graph should be 2B/%g. Thus this slope can be used to calculate an experimental value for g, which is then compared to the theoretical value of 9.80 m/s2.2

1

Douglas C. Giancoli, Physics, Prentice-Hall, 1998, p. 319

2

Ibid., p. 34
1

Part I: Dependence of Period on Mass
The length (L) of the pendulum is held constant at 30 cm while the mass (m) is varied from 50 g to 500 g, in 50 g increments.
The time required for 30 cycles is measured in three trials and represented by t1, t2, t3. The period (time for one cycle) is represented by T and is calculated from the average value of t: T = tavg/30. All times are measured in seconds.

The mass is measured in grams, then converted to kilograms.

Table 1: Period vs. Mass for Simple Pendulum
t1 (s)

t2 (s)

t3 (s)

tavg (s)

T (s)

m (g)

m (kg)

32.57

32.51

32.53

32.54

1.085

50

.050

32.85

32.90

32.88

32.88

1.096

100

.100

33.01

33.01

33.00

33.01

1.100

150

.150

33.06

33.01

33.06

33.04

1.101

200

.200

32.96

33.11

33.04

33.04

1.101

250

.250

32.98

32.99

33.00

32.99

1.100

300

.300

32.96

32.96

32.96

32.96

1.099

350

.350

32.96

32.96

32.94

32.95

1.098

400

.400

32.94

32.99

32.99

32.97

1.099

450

.450

32.86

32.83

32.83

32.84

1.065

500

.500

The data for T vs. m from this table is represented on the graph on page 3*. The graph on page 4 shows the same data with an exaggerated scale for the period. We see that the variation in the period is no more than 0.036 s (3.29%).

*All graphs and slope and uncertainty calculations for this report were done using QuatroPro 2

Period vs. Mass for Simple Pendulum

Period (s)

2

1

0
0

0.1

0.2
0.3
Mass (kg)

0.4

0.5

3

Period (s)

Period vs. Mass for Simple Pendulum
(Exaggerated Scale)

1.08
0

0.1

0.2
0.3
Mass (kg)

0.4

0.5

4

Part II: Dependence of Period on Length
The mass (m) is held constant at 250 g and the length (L) is varied from 10 cm to 50 cm, in 5 cm increments..
Times are measured and period calculated as in part I. As before, all times are in seconds. Lengths are measured in centimeters, then converted to m.

Table 2: Period vs. Length for...