The Simple PendulumThe Simple Pendulum

Recall from lecture that a Recall from lecture that a

pendulum will execute simple pendulum will execute simple

harmonic motion for small harmonic motion for small

amplitude amplitude vibrations. vibrations.

Period (T) - time to make one Period (T) - time to make one

oscillationoscillation

Frequency (f) - number of Frequency (f) - number of

oscillations per unit timeoscillations per unit time

secvibration/21

FrequencyFrequency PeriodPeriod

ionsec/ vibrat21

ionsec/ vibrat31

ionsec/ vibrat41

ionsec/ vibrat2

ionsec/ vibrat1

/ secvibrations3

secvibration/1

/ secvibrations4

/ secvibrations2

FrequencyPeriod 1

fT 1

In symbolic formIn symbolic form

oror

The period is independent of the mass of the The period is independent of the mass of the

pendulum.pendulum.

The period depends on the length of pendulum.The period depends on the length of pendulum.

It also depends on the amplitude (angle of It also depends on the amplitude (angle of

swing). swing).

If the displacement angle is small (less than 10If the displacement angle is small (less than 1000),),

then the period of the pendulum depends then the period of the pendulum depends

primarily on the length (primarily on the length (l l ) and the acceleration ) and the acceleration

due to gravity (due to gravity (gg) as follows.) as follows.

g2T l

It must be emphasized again that this equation is goodIt must be emphasized again that this equation is goodfor small angles of vibration but not for large.for small angles of vibration but not for large.

Squaring both sides of the equation Squaring both sides of the equation yieldsyields

Let’s rewrite this equation to getLet’s rewrite this equation to get

l

g

4T2

2 π

g22 4T l

T T 22 is is yy

4422/g/g is is mm

ll is is xx

and and bb will equal zero will equal zero

l

g

4T2

2 π

This is of the form (from last week’s This is of the form (from last week’s lab)lab) mxy b

Therefore by plotting Therefore by plotting T T 22 versus versus ll and and using the slope of this curve one can using the slope of this curve one can determine the acceleration due to determine the acceleration due to gravity gravity gg. The slope is. The slope is

g4

slope2π

Multiply both sides Multiply both sides of the equation by of the equation by g g and getand get (g)g

4slope(g)

2π

24slope(g) πThis reduces toThis reduces to

Now divide both Now divide both sides by the sides by the slopeslope to get to get

which reduces to which reduces to slope4

slopeslope

(g)2π

slope4g

2π

g4

slope2π

Purpose of Today’s ExperimentPurpose of Today’s Experiment

You will determine the local value of You will determine the local value of the acceleration due to gravity by the acceleration due to gravity by studying the motion of a simple studying the motion of a simple pendulum.pendulum. Note: Pendulums are used in a variety Note: Pendulums are used in a variety of applications from timing devices like of applications from timing devices like clocks and metronomes to oil clocks and metronomes to oil prospecting devices. prospecting devices.