Two important examples of s.h.m. © D Hoult 2010

The mass / spring oscillator

The mass / spring oscillator

net force acting on the mass = zero

A graph of force against extension for a spring is a straight line passing through the origin: force a extension

A graph of force against extension for a spring is a straight line passing through the origin: force a extension The constant of proportionality, k is called the elastic constant of the spring (usually considered to be a positive number)

When the displacement is downwards, the net force is upwards and vice versa

So a better way to draw the graph would be

In this case, F represents the force with which the spring “pulls back” when it is given an extension x by some external agency

slope = - k

F = - k x

F = - k x If the mass is released after having been given a displacement, x, then its acceleration will be a =

F = - k x If the mass is released after having been given a displacement, x, then its acceleration will be - k x a = m

F = - k x If the mass is released after having been given a displacement, x, then its acceleration will be - k x a = m So the motion is s.h.m. and the constant of proportionality between a and x has magnitude

F = - k x If the mass is released after having been given a displacement, x, then its acceleration will be - k x a = m So the motion is s.h.m. and the constant of proportionality between a and x has magnitude k m

F = - k x If the mass is released after having been given a displacement, x, then its acceleration will be - k x a = m So the motion is s.h.m. and the constant of proportionality between a and x has magnitude k m which is equal to w2 in the “shm equation”

Remembering that 2p T = w we can suggest that the time period of a mass spring oscillator is given by

Remembering that 2p T = w we can suggest that the time period of a mass spring oscillator is given by m T = 2p k

The simple pendulum

F =

F = mg sin q

F =

F = - mg sin q

a =

a = - g sin q

q =

x q = L

If we make sure that q is a small angle

If we make sure that q is a small angle, sin q can be replaced by q

g a = - x L