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EM.ES.5 Vibrations Lab. تازازتهلاا ربتخم 5 Experiment No. (1) SIMPLE PENDULUM 1.1 Objective (1) To determine the magnitude of gravitational constant (g). (2) Determine the natural frequency of oscillation of system. 1.2 Introduction One of the simplest examples of free vibration with negligible damping is the simple pendulum. The motion is simple harmonic. 1.3 Theory of Experiment Figure ( 1.1 ) T = Tension in wire. m= mass of ball. = wire length. 𝜃 mg m m 𝑇 𝑠𝑖? 𝜃 𝑇 ??𝑠 𝜃 𝜃 x m 𝑑 2 𝑥 𝑑𝑡 2 ?
acceleration. g= gravitational constant. t= periodic time. ( ) (P.E) = Potential energy. Displace mass by small angle ( 10 ) and let free resolve forces (use newton's second law) external forces in horizontal direction = mass * acceleration = m 2 2 .……….(1.1) external forces in vertical direction = 0 0 …….…..(1.2) Since( ) small ; and =1 and in eqt's (1)&(2) and sub. For (T) from (1.2) into(1.1) will obtain eqt. of motion. m 2 2 + m g = 0 …………..(1.3) 2 2 = 0 …………..(1.4) Since harmonic motion , frequency of ; ..….……..(1.5) Periodic time t = = …………(1.6) Or can use energy method to obtain eqt. of motion . (K.E) Kinetic energy = ̇
(P.E) Potential energy = mg( ) Use Rayleigh principle or rate change of momentum ( ) 0 m ̈ + mgL = 0 m ̈ + mgL 0 m ̈ + mg L = 0 ̈ = 0 In this experiment, the object is to analyze the above equation for the periodic time by varying the length of the pendulum and timing the oscillations. The independence of the size of the mass of the particles is demonstrated. 1.4 Apparatus Figure (1.2) shows the following:- Sub-frame (cross beam) (B1) Small wooden ball ( B2) Small steel ball ( B3) 1. Inextensible flexible cord 2. Stopwatch or clock 3. Meter rule Both the steel and the wooden balls attach to lengths of cord approximately one meter long, each of the two cords suspending from the small chucks at either end of the sub-frame. You can vary the length by pulling the thread through the chuck and the hole above the sub-frame.
Sub-frame B1 Frame Figure (1.2 ) Apparatus of Experiment 1.5 Procedure Measure and note the length L ,( the distance from the bottom of the chuck to center of the ball). Displace the pendulum through a small angle and allow swinging freely. Once settled measure the time taken for 50 oscillations and record the periodic time, t . Repeat the procedure for various values of ( L ) for both the wooden ball and the steel ball. Enter the result in table (1.1). Plot a graph for values of against values of length L . 1.6 Results Tablet (1.1) Experiment No. 1 Results Length L (m) Time for 50 complete oscillations Period t one oscillation Steel wood steel wood steel wood Inextensible wire Steel ball B3 B2 Wooden ball θ θ