Lesson 23 notes – Defining SHM
Objectives
Be able to define simple harmonic motion;
Be able to select and apply the equation a = – (2πf)2 x as the defining equation of
simple harmonic motion;
Be able to select and use x = Acos(2πft) or x = Asin(2πft) as solutions to the
equation a = – (2πf)2 x.
Outcomes
Be able to define simple harmonic motion.
Be able to select and apply the equation a = – (2πf)2 x as the defining equation of
simple harmonic motion.
Be able to select and use x = Acos(2πft) or x = Asin(2πft) as solutions to the
equation a = – (2πf)2 x.
Be able to explain why the period of an object with simple harmonic motion is
independent of its amplitude.
positive
The restoring force in SHM
Imagine a mass on a spring being displaced and then dropping through the
equilibrium position to as far down as it can go and back up again:
If we let the mass hang it will stay in the equilibrium point. The strain of the spring
equals the weight of the mass.
If we pull the mass up, the strain is less and so it will accelerate towards the
equilibrium point since gravity is still the same.
If we pull it down the strain will be greater than the weight and so it will accelerate
towards the centre again.
At the equilibrium point the forces cancel leaving no acceleration and a maximum
velocity.
If we consider up positive then any displacement, velocity or acceleration
directed downwards will be negative.
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So displacement up gives a resultant force (and hence acceleration) down, and
vice versa. We call this force the restoring force (because it ties to restore the
mass to its equilibrium position). The greater the displacement, the greater the
restoring force.
We can write this mathematically:
Restoring force F  displacement x
Since the force is always directed towards the equilibrium position, we can say:
F -x
or
F = - kx
Where the minus sign indicates that force and displacement are in opposite
directions, and k is a constant (a characteristic of the system).
This is the necessary condition for SHM.
Now since we are dealing with vector quantities here it makes sense to use a
sign convention. We call the midpoint zero; any quantity directed to the right is
positive, to the left is negative. (For a vertical oscillation, upwards is positive.)
Think about mass-spring systems. Why might we expect a restoring force that is
proportional to displacement? (This is a consequence of Hooke’s law.)
Equations of SHM
SHM can be represented by equations. For displacement:
x = A sin 2ft
or
x = A sin t
f is the frequency of the oscillation, and is related to the period T by f = 1/T. The
amplitude of the oscillation is A.
x = A cos 2ft is also a possible solution depending on where the
oscillation starts.
Velocity: v = 2f A cos 2ft
or
v= A cos t
2
Acceleration: a = - (2f)2 A sin 2ft
or
a = -2 A sin t
Comparing the equations for displacement and acceleration gives:
a = - 2x
and applying Newton’s second law gives:
F = - m 2x
These are the fundamental conditions that must be met if a mass is to oscillate
with SHM.
If, for any system, we can show that F  -x then we have shown that it will
execute SHM,
and its frequency will be given by:

 = 2f
so  is related to the restoring force per unit mass per unit displacement.
Extension
Developing equations by differentiation.
For those who like a challenge – or those studying maths you should be able to show that
differentiation of: x = A sin 2ft twice will get you to: a = - (2f)2 A sin 2ft.
Have a go! Next lesson will show you how to solve the equations graphically and
this will also help your understanding.
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