Experiment 15

Experiment 15: Simple Pendulum

Purpose

(1) To observe the motion of a Simple Pendulum.

(2) To determine the acceleration of gravity g.

Apparatus

Simple pendulum, a meter stick, a timer, a dial caliper.

Theory

The simple pendulum consists of a small heavy sphere of mass m suspended by a string of length L, with the mass of the string negligible compared to m.

If the string is displaced an angle a from the vertical (see figure) and released, then it will execute a periodic motion between points A and B. If the angle a is small, this motion is accurately described by Simple Harmonic Motion with the Formula 1:

where T is the period and g is the acceleration due to gravity.

Procedure:

1. Set the length of the pendulum L to about 80 cm. Align and adjust the pendulum so that, with a as large as 500 , it is able to make more than 30 swings:

(a) without hitting any supports

(b) without any changes of its suspension point

(c) without swinging appreciably out of the vertical plane

2. Lengthen L to be close to 100 cm and measure it to 1 mm accuracy from the point of suspension to the middle of the sphere. Record the length as L1. Set the pendulum to oscillate by releasing it from a very small angle a (no greater than 50 ). Measure and record the time TN needed to make N full oscillations. Do this three times, for N = 50, 70 and 90.

Table 1. L1 = ….
N / TN
(units are?) / T1
(units are?)
Average <T1

3. Shorten the string to about 90 cm and record the length as L2. Repeat (2) above to find <T2>.

4. Shorten the string to about 80 cm and record the length as L3. Repeat (2) above to find <T3>.

5. Using L3, set the pendulum to oscillate 30 times with an angle larger than 450 , and record the time as TLargeAngle.

Lab Report

1. In Table 1 calculate T1 and <T1> to three significant digits.

2. Fill out similar tables for L2 and L3.

3. Using Formula 1 solve for g for each length . Show your equation and a sample calculation including units. Find the % discrepancy between your average value <g> and the correct value

g = 980 cm/sec2.

Calculation of g
L / <T> / g
Average <g

4. From the data in step 5, calculate the period of oscillations T’. Display the % difference between TLargeAngle and <T3> for L3. What is the main reason for this difference?

Question #1 What is the reason for measuring the time with three different values of N rather than repeating it three times with the same value of N?

Question #2[*] Suppose that the value of T is measured exactly, but L is off by 1%. How much will g be off in % terms?

Question #3[*] Suppose that the value of L is measured exactly, but T is off by 1%. How much will g be off in % terms?

74

[*] Students with a working knowledge of calculus should be able to answer Questions #2 and #3 by using differentials to represent small differences. But all students should be able to handle these questions by quick arithmetic using calculators.

[*]