13.1.1 simple harmonic motion

Part 2: translating circular motion to simple harmonic

Simple harmonic motion

Linear motion: a - constant in size + direction Circular motion: a - constant in size only Oscillatory motion: a changes periodically in

size + direction (like x and ) SHM is a special form of oscillating motion Pendulums and masses on springs exhibit

SHM A body oscillates with SHM if the

displacement changes sinusodially

Linking circular motion and SHM The arrangement shown below can be used to

demonstrate the link between circular motion and SHM

Adjusting the speed of the turntable will allow both shadows to have the same motion (i.e. move in phase)

The shadows are the components of each motion parallel to the screen and are sinusoidal

Let N represent the sinusoidal motion of one shadow

It oscillates about O (equilibrium point) in a straight line between A and B

O

NA B

Displacement and velocity When N is left of O:

- x is left- is left when moving away from O and right when moving towards O

When N is right of O:- x is right- is right when moving away from O and left when moving towards O

The size of the restoring force increases with x BUT always acts towards equilibrium point (O)

F -x resulting acceleration must behave

likewise, since F = ma and m is constanti.e. a increases with x but acts towards O

a -x In oscillations a and x always have

opposite signs

Definition

“If the acceleration of a body is directly proportional to the distance from a fixed point, and is always directed toward that point, then the motion is simple harmonic”

a -x or a = -(+ve constant) x Many mechanical oscillations are nearly simple

harmonic, especially at small amplitudes Any system obeying Hooke’s law will exhibit SHM

when vibrating

Equations of SHM

Consider the ball rotating on the turntable The ball moves in a circle of radius r It has uniform angular velocity The speed, v around the circumference

will be constant and equal to r (v = r) At time t the ball (and hence the bob of

the pendulum) are in the positions shown:

Displacement

Angle = t (since = /t) The displacement of the ball along OF from

O is given by: x = r cos = r cos t For the pendulum (and masses on springs),

the radius of the circle is equal to the amplitude of its oscillation i.e. r = A

Hence x = A cos t But = 2f x = A cos 2f t

r

x

Velocity The velocity of the pendulum bob is equal

to the component of the ball’s velocity parallel to the screen (i.e. along y-axis)

Bob Ball

v = r

O

r

velocity = -v sin

Ball Bob

Velocity of bob = -v sin = -r sin = /t = t Velocity of pendulum bob, = -r sin t

Sin is +ve when 0° 180° (i.e. bob or ball moving down)

Sin is –ve when 180° 360° (bob or ball moving up)

Negative sign ensures velocity is negative when moving down and positive when moving up!

Variation of velocity with displacement

sin2 + cos2 = 1 sin = (1- cos2) = ± r sin = ± r (1- cos2) From earlier x = r cos , so x/r = cos (x/r)2 = cos2

By substituting: Velocity = ± r (1- cos2)

= ± r (1 - (x/r)2)= ± (r2 – x2)

Recall = 2f Hence velocity of pendulum in SHM of

amplitude A is given by: = ± 2f (A2 – x2)

Acceleration The acceleration of the bob is equal to the

component of the acceleration of the ball parallel to the screen

The acceleration of the ball, a = 2r towards O

So the component of a along OF = 2r cos

Bob Ball

a = 2/r a = 2r cos

O

O

Hence, the acceleration of the bob is given by: a = -2r cos (-ve since moving down)

Since x = r cos and = 2f a = -(2f)2 x Since (2f)2 or 2 is a +ve constant, equation

states that acceleration of the bob towards the equilibrium point O is proportional to the displacement x from O

The acceleration is zero at O and maximum when the bob is at the limits of its motion when the direction and motion changes

Time period Period T is time taken for the bob to

complete one oscillation In the same time the ball has made one

revolution of the turntable T = circumference of circle

speed of ball T = 2r

Since = r T = 2 For a particular SHM is constant and

independent of the amplitude (or radius) of the oscillation

If the amplitude increases, the body travels faster T is unchanged

A motion with constant T, whatever the amplitude, is isochronous - this is an important characteristic of SHM

Time traces of SHM Displacement

T/4 T/2 3T/4 T

Note: the gradient = velocity

Velocity

T/4 T/2 3T/4 T

Note: when = 0, a = max

Acceleration

T/4 T/2 3T/4 T

a = 0 when = max

All graphs are sinusoidal When the velocity is zero, the acceleration

is a maximum and vice versa There is a phase difference between

them Between and a phase difference = T/4 Between x and a phase difference = T/2

Summary: equations of SHM

Frequency f = /2 Period T = 2/ Displacement x = A cos t

= A cos 2ft Velocity = A2 – x2

= 2f A2 – x2

Acceleration a = -(2f)2 x