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## The Simple Pendulum

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**Learning Objectives**Book Reference : Pages 42 The Simple Pendulum To establish which factors influence the period of a pendulum To understand how the period of a simple pendulum can be used to establish a value for g experimentally**During the practical last lesson, we investigated three**factors which may have an effect upon the period of a simple pendulum: • Mass of the bob • Length of the string • Initial displacement (or amplitude) • Which factors did you find affected the period? • [VPL : SHM] Factors Which Affect the Period**Consider a simple pendulum with a bob of mass m suspended by**a thread of length L. Which has a displacement s from the equilibrium position giving an angle to the vertical Period of a Simple Pendulum 1 L s**The weight of the bob (mg) has the following components:**• Perpendicular to motion : mg cos • Parallel to motion : mg sin • The restoring force F causing the SHM will be in the opposite direction: • F = -mg sin (Using F=ma for a) Period of a Simple Pendulum 2**a = F/m = (-mg sin ) / m**• a = -g sin • For small values of , (< 10°), sin = s/L • a = -g s/L • and we know that: • Acceleration = - (2f)2 x displacement • a= - (2f)2 s Period of a Simple Pendulum 3**a = -g s/L = - (2f)2 s (remove s and -)**• g /L = (2f)2 • f = (g /L) / 2 (since T = 1/f) • T = 2 (L /g) Period of a Simple Pendulum 3**From the equation we can see that:**• The period is independent of mass • The period is independent of initial displacement (amplitude) • The period is dependent upon the length of string • More specifically T2 is proportional to L & and a graph of T2 against L will have a gradient of 42/g • This can be used to establish a value for g Period of a Simple Pendulum 4**As the bob passes through the equilibrium position....**• The tension acts directly upwards and provides a centripetal force • Tension – mg = mv2/L Period of a Simple Pendulum 5**This work was first carried out in 1581 by Galileo when he**observed lamps swinging backwards and forwards in the cathedral at Pisa Period of a Simple Pendulum Since clocks had not yet been invented he used his own pulse for timing!**Having conducted the experiment how could it be improved?Why**are the pendulum bobs in clocks not spherical? Improving the experiment**Calculate the time period of a simple pendulum with lengths**• 1.0m • 0.25m • Take g to be 9.81 m/s2 • Now calculate the period for the 1.0m pendulum on the moon where gravity is around 1/6th of that on Earth (g=1.6 m/s2) Problems 1