13.1.1 simple harmonic motion Part 2: translating circular motion to simple harmonic
May 25, 2015
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