‘Simple Harmonic Motion’ is quite a complicated term for something which is on the surface, relatively simple. You see simple harmonic motion all around you in the world every day. Clock pendulums, children’s swings, roundabouts on parks a bicycle wheel, anything that moves in a cycle.
The way it was taught to me in college is in terms of a pendulum, if you imagine a clocks pendulum it swings from one point through the middle of the swing (the origin, or equilibrium point’) reaching a point where it stops, and then swings back in the other direction through the equilibrium and back to where it began, this is one oscillation and the time it takes to complete this is called the ‘Period’ (denoted ‘T’)
in basic SHM (assuming there is nothing slowing down or ‘damping’ the movement of the pendulum (meaning it will move at constant velocity forever, so it is an unrealistic model) the equation for the period of one swing is:
T = 2 x (Pi) x Square root (Legnth of string/acceleration due to gravity)
this applies to all pendulums and the acceleration due to gravity (denoted ‘g’) is 9.81 m/s/s on earth so with only the legth of the string you can work out the exact time it takes the pendulum to do one complete period.
Frequency is very similar to period, where period is how long it takes to do one oscillation, frequency is the number of oscilliations which occur in one second. It is measured in Hertz and has the units of ‘per second’
The relationship between period (T) and frequency (f) is very simple: T = 1/f which can also be arranged to; Tf = 1 and f = 1/T
Ok, now onto the harder stuff, When an object moves in a circular path it is in-fact accelerating away from a central point (the centre of the circle) this is due to the centripetal force you come across this most obviously on spinning fairground rides, you feel as if you are being pushed into your seat as you spin faster, yet in-fact you are being pushed away by the motion and the force you are feeling is the seat exerting an equal but opposite force back onto you.
So, anything in SHM (simple harmonic motion) is in-fact accelerating, which also means it has a velocity which is changing, and this acceleration is due to the unbalanced force thats acts around the centre point (in this case the end of the pendulum string). The acceleration of the pendulum at the equilibrium point is in-fact zero. At these same points, where displacement from the origin is also zero, velocity is at its maximum value.
Velocity (denoted ‘V’) of the pendulum has a more complex equation:
V = [2(Pi) x f] x square root (Amplitude^2 – Displacement ^2)
Amplitude (denoted ‘A’) is the maximum distance that the pendulum can reach from the origin before swinging back to the origin (can be positive and negative in different directions) and displacement is the distance from the origin at the time you want to find the velocity. so if you want to find what the maximum velocity is, this is at the equilibrium and displacement (denoted ‘X’) is zero, and so the equation can be simplified to…
Vmax = 2(Pi)fA
At the maximum points of the pendulums swing (the amplitude) Velocity is in fact zero, as the amplitude squared minus displacement squared in the final brackets cancel out. At these points interestingly acceleration is maximum, despite the pendulum being still. This is because if is accelerating from one direction to another, to pull the pendulum back to equilibrium. The equation for acceleration of a pendulum is as follows:
a = – [2(Pi)f]squared x (X)
where ‘a’ is acceleration, ‘f’ is frequency and ‘X’ is displacement.
The final complex equation is to show the displacement ‘X’ from the origin at time ‘t’:
X = A cos [2(pi)f] x t
where ‘A’ is amplitude, ‘f’ is frequency and ‘cos’ is the cosine function.
Finally, in a pendulum we can calculate the total energy of the system this is the total of Kinetic energy (KE) and Potential energy (PE) at any given point. The easiest way to do this is to find what Vmax is using the equation above, then squaring the answer to put into:
KE = (1/2)mV^2
Where ‘m’ is the mass on the end of the pendulum and ‘V’ is the velocity. At this point potential energy is zero and as such this solution gives the total energy of the system. Likewise when KE = 0 PE is at its maximum.
Another common example of SHM is a mass and spring system, interestingly these systems are not effected by gravity, a mass on a spring bobbing up and down on the surface of mars will have the same period and frequency and the same system here on earth. Its is dependant on the spring constant (k) and the size of the mass on the spring (m) and is given by:
T = 2(Pi) x square root(m/k)
in this system other useful equations are:
a = – (k/m)X
and
mg = ke
where; ‘k’ is spring constant, ‘e’ is extension caused by the mass, ‘m’ is the mass and ‘g’ is the size of the gravitational force.
