Simple Pendulum. L Length of pendulum; this is from the bottom of the pendulum support to center of mass of the bob.

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1 Simple Pendulum Many mechanical systems exhibit motion that is periodic. Generally, this is because the system has been displaced from an equilibrium position and is subject to a restoring force. When this restoring force is proportional to the displacement from equilibrium, we say that the motion is simple harmonic (SHM). Theory For the following definitions, reference Figure 1 and Equation 1. m Mass of pendulum bob. L Length of pendulum; this is from the bottom of the pendulum support to center of mass of the bob. θ Displacement from equilibrium - also known as the amplitude. Notice that this displacement occurs on both sides of the equilibrium position. T The period of oscillation of the pendulum (the time for one complete oscillation). From amplitude on one side of equilibrium, over, then back is one oscillation. Figure 1: A Simple Pendulum When we refer to a simple pendulum, it is understood that the mass (bob) that is oscillating is much greater than the mass of the support (string). Provided that the amplitude is kept small (θ < 15 ), the motion of 1

2 the pendulum approximates SHM. The motion is not truly SHM since the restoring force is proportional not to the displacement θ, but to the sine of θ. However, for small amplitudes, it can be shown that the period of the simple pendulum is given by T = 2π where g is the gravitational acceleration. Note that the period of a simple pendulum is dependent only upon its length. L g (1) Apparatus Table clamp, Vertical rod, Pendulum clamp, String, Bobs of varying mass, Protractor, Meter stick, Stopwatch, Balance. Procedure In all procedures you should use a constant (and relatively small; θ < 15 ) initial amplitude to set the pendulum into motion. Make sure that you do not impart a tangential velocity to the bob as you release it - the oscillations should ideally be confined to a vertical plane. Rather than trying to time one oscillation for the experimental period T of the pendulum, time a larger number of oscillations. To determine the period, simply divide this total time by the number of oscillations. Mass and Period 1. Experimentally determine the period of oscillation for bobs of varying mass (everything else remains constant). Place all data in Table 1. Length and Period 1. Experimentally determine the period of oscillation for varying lengths of the pendulum (everything else remains constant). Place this data in Table Graph the Experimental Period Squared vs. the Length. Plot the straight line of best fit over the data points and determine the slope of the line. 2

3 Table 1: Period and Mass Initial amplitude ( ) Pendulum length (m) Number of oscillations timed Mass Total Experimental Time Period (g) (s) (s) 3

4 Table 2: Period and Length Initial amplitude ( ) Bob mass (g) Number of oscillations timed Pendulum Total Experimental Experimental Length Time Period Period Squared (m) (s) (s) (s 2 ) Analysis Mass and Period 1. Calculate the average period as well as the average deviation in the period. What is the deviation expressed as a percentage of the average? 4

5 2. According to your answers in Question 1, what affect did the mass of the bob have on the period of the pendulum? Is this consistent with Equation 1? If so, how? If not, why not? Length and Period 1. What relationship did you test between period and length? Does your graph verify this relationship? Explain why or why not. 2. Use the slope of the line on the graph you plotted to calculate an experimental value of g. Show all work. 5

6 Pre-Lab: Simple Pendulum Name Section Answer the questions at the bottom of this sheet, below the line - continue on the back if you need more room. Any calculations should be shown in full. 1. What is amplitude? 2. What is the period of a pendulum? 3. A simple pendulum has a length L of 55.0cm. What is its theoretical period? 4. You time 20 oscillations of the pendulum in Question 3; the total time is 29.50s. What is the experimental period? 5. According to Equation 1, what type of curve would you expect if you were to plot Period vs. Length? Provide a rough sketch. 6