SHM and Circular Motion

Simple Harmonic Motion and Circular Motion are very closely related.  Think about a turntable with a spike attached to it.  Its shadow is projected onto a screen.  As we turn the turntable, we can see the shadow moving forwards and backwards on the screen

If we look at the apparatus from above:

As P goes round the circumference of the revolving turntable, its projection N will move up the diameter of the circle AOB.  The shadow of P at this instant is projected onto the screen at N´.  The shadow will move up and down the screen along the line A´N´O´B´.  The turntable is revolving at an angular velocity of w radians per second, and its linear speed is wr ms-1.

How can we show that the image of P describes simple harmonic motion?

Let us consider our shadow going across the screen:

We know that the acceleration is towards the centre of the circle and is given by a = (2pf )2r.

Acceleration is a vector, so has horizontal and vertical components.  We can work out the acceleration parallel to ONA as:

                                                            a = (2pf )2 r cos q

  Since the acceleration is towards the centre, it is given a minus sign, so our formula is modified to:

                                                            a = -(2pf )2 r cos q

  If we look at the distance from the equilibrium point, we can calculate it as:

                                                            x = r cos q

So we can combine these two to give:

                                                            a = - (2pf )2 x

  This satisfies the condition for SHM that a = -Kx; in this case K = (2pf )2. A useful little dodge here is that p2 » 10.

The time it takes for our turntable to make one complete revolution is called the period, and is given the code T, and is measured in seconds.  It is also the time for the shadow of P to make one oscillation, or complete to-and-fro movement.  We can use the simple equation

                                                time (s)  = distance (m)

                                                                  speed (m/s)

 and

                                                period (s) = circumference of the turntable (m)

                                                                                     linear speed (m/s)

 

                                                T = 2pr = 2pr = 2p

                                                         v       wr       w

  Now we know that frequency, f = 1/T Þ f = w  Þ w = 2pf

2p                                           

  In some texts you may see the equation for acceleration in SHM written as:

                                                a = - w2 x

T is independent of the radius of the turntable, hence the amplitude of the oscillation.  If the amplitude is increased, the body travels faster, so T is not affected.  This kind of oscillation is called isochronous, which means that it takes the same time to complete each cycle.

The direction of the velocity of anything moving in a circle is always at a tangent:

So the component of the velocity parallel to AOB is:

Since v = (2pf )r, the velocity parallel to AOB = -v sin q = -(2pf )r sin q

The negative sign tells us that the velocity is negative when the image is going upwards and positive when the image is going downwards.  This ties in with the fact that the sine function has positive values for values of q between 0 and p radians (0 – 180 o) and negative values from p to 2p radians (180 o – 360 o).

The derivation of the equation that gives us velocity at any point in the oscillation is rather tedious, but the relationship is:

v2 = (2pf )2(A2 – x2)

Þ v2 = 4p2f 2(A2 – x2)  

Þ v = 2pf Ö(A2 – x2)

In this relationship, A is the amplitude and s is the displacement from the equilibrium position.  If x = 0, v has a maximum value; if x = A, v = 0.  The velocity is 0 at each extreme of the oscillation.

We can easily find the displacement using our circular argument.  If the radius of the turntable is r, we can show quite simply that the displacement, s, if given by:

  x = r cos q

Since q = wt, we can rewrite this as:

s = r cos wt

  The radius of the circle is also the amplitude, and w = 2pf, we can rewrite this as:

                                                                     s =  ± A cos 2pft

The plus and minus sign here tells us that the motion is forwards and backwards.  Which sign we give for direction is up to the individual.  Generally left to right is forwards.

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