Simple+Harmonic+Motion

=Simple Harmonic Motion (SHM)=

Defining SHM
In simple harmonic motion the displacement of the motion is directly proportional to the force causing it, but the force acts in the opposite direction to the displacement. Typical examples of SHM are a swinging pendulum, or a mass oscillating on a spring.

This can be represented by the equation...F = - ky

where F is the force in Newtons, k is the force constant, y is the displacement in metres and the negative sign identifies that the force acts in the opposite direction to the displacement.

From Newton's Laws we know that acceleration is directly proportional to the net force causing it, so it must follow that acceleration must also be directly proportional to displacement, but in the opposite direction. This is represented by the equation.... a = - w^2 y

where a is acceleration in m/s^2, w is angular frequency in rad/s and y is displacement in m.

Solving SHM problems
Typical SHM problems can be solved in one of two ways;

EITHER using formula such as y = Asinwt, v = wAcoswt, a = -w^2 Asinwt

If using formula you must determine whether timing starts at maximum displacement or at zero displacement. You must also remember to have your calculator in radians, and for some reason the formula writers neglect to put in brackets ie the formula y = Asinwt really should be written y = Asin(wt)

OR using reference circles.

These are used to find displacement at a given time or the time taken to get to a particular displacement. For both these situations you need the period, T of the motion and the amplitude, A.

It is important to get the set up of the reference circle correct. You must first determine whether the motion is horizontal or vertical. Then you must identify the starting position and place this on the circle.

If using the reference circle to solve a problem relating to displacement, then you need to estimate the position for the given time, t, ie work out which quarter of the circle it will be in given by relating the time to the total time for period. You can calculate this angle using t/T x 360. Once you have the angle you can construct a right angle triangle on your reference circle. The hypotenuse will be equal to amplitude, A. then you can calculate the unknown side of your triangle to work out displacement and subsqunetly answer the question.

If using the reference circle to solve a problem relating to time taken, then you need to mark the destination point on your reference circle and construct a right angled triangle to work out the angle, theta, moved through to get to this destination. Convert this into time using theta/360 x T.