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The Simple Pendulum. 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.

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the simple pendulum1

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

factors which affect the period

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

period of a simple pendulum 1

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

period of a simple pendulum 2

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

period of a simple pendulum 3

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 = - (2f)2 x displacement
        • a= - (2f)2 s

Period of a Simple Pendulum 3

period of a simple pendulum 31

a = -g s/L = - (2f)2 s (remove s and -)

  • g /L = (2f)2
  • f = (g /L) / 2 (since T = 1/f)
  • T = 2 (L /g)

Period of a Simple Pendulum 3

period of a simple pendulum 4

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 42/g
  • This can be used to establish a value for g

Period of a Simple Pendulum 4

period of a simple pendulum 5

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

period of a simple pendulum

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!

problems 1

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