1 0f 3 Simple Pendulum Pu rp 050 1. To study the simple pendulum and how the period time, T relate to the Length, L of the simple pendulum. 2. To find the value of g, the acceleration of gravity near the earth surface, based on the relation between the period, T and the length, L of the simple pendulum. Introduction A simple harmonic motion is a motion where the net force, F on an object, called the simple harmonic oscillator, has a magnitude that is proportional to the displacement from some equilibrium position, Er. and a direction that always points towards the equilibrium position and is therefore known as a "restoring\" force: restoring = C' E; (1) where c is a constant. For simplifying the notation we will write EE simply as x, where it is understood that x is the displacement from some equilibrium position. Displacement and acceleration of a simple harmonic oscillator are related by the equation: a = wzx (2) This last equation is equivalent to the statement of SHM in Eq. {1), since force and acceleration are related by Newton's 2"'1 Law. to is called the angular speed of the motion. For simple harmonic motion a; is constant and is given by: 2rr . M? (3) : where T is the time of one complete oscillation for example, a simple pendulum is as shown in figure 1. It is a mass {called a bob) attached to a string. The mass of the string is negligible compared to the mass of the bob. The time it takes for the mass of the simple pendulum {the bob] to oscillate from maximum displacement at the right side ail the way to the maximum displacement at the left side and return back to the maximum displacement at the right side is T, called the period. mg sin 3 Consider again figure 1. For the simple pendulum, mg Fm, = Preston-Hg = mgsin6 where 9 is the angle made by the string to Figure 1: A simple pendulum. (From the vertical. The negative sign is again because Preston-mg opposes any Douglas c, Giancoli, Physi'cs; Principfes increase in 9. The motion of a simple pendulum is simple harmonic in the With Applications. 51" Edition, Ch. 11. limit the mass of the string is negligible compared to the mass of the Pear-"0\" Prentice \"3":2005) pendulum bob {the metal sphere attached to the string}, and that the string does not stretch {inextensible}. For a small displacement angle, 6, -mgsin6 z mgl. But F = ma {Newton's 2"d Law), and using Eq. {2} above and Fig. 1, we get: mg 6 = mwzx (4-) Data Sheet Name: Group: Date experiment performed: Steps (1) to (3): m = 1kg, 0. = 50 Table L (m) Tcalculated (s) using eqn. (5) experimental Time of 5 Period, Texperiemntal (S) Texperimental ($2) cycles, t (s) (using simulator stop watch) 0.5 0.6 0.7 0.8 0.9 1.0 Slope of Texperimental Versus L: calculated g from slope: % error: Step 4) m = 1kg, L = 1m, 0. = 10: Answer to a) Answer to b) Step 5) m = 0.5 kg, L = 1m, 0. = 10 Answer to a) Answer to b) Answers to Questions: 1. 2