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 gL / <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?
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[*] 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.
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