Simple Harmonic motion

F=ma , why is acceleration of a simple Harmonic motion proportional to displacement?

This isn't going to be as helpful as we would wish but it's a start.

Since we know that in SHO they have a displacement that looks like a sine wave. If we differentiate this twice, you get -sine.

This shows that the acceleration (since this is always a double differentiation of displacement) is proportional to displacement because f(x) is proportional to k*f(x)



I'm going to try to think of a more convincing explanation and see what others say. hF!

Because the resultant force on the object (often called the restoring force in this context, because it acts to try to return the object to its rest position) is proportional to the displacement of the object from its rest position. This isn't really explaining anything (it is just using F = ma to say the same thing), but is a bit easier to see. Think of a mass attached to an open-wound spring, moving on a smooth horizontal surface, with the other end of the spring fixed. The more the spring extends, the larger the tension in the spring - and you know from Hooke's law that this tension is proportional to the extension of the spring, which in this case is the displacement from the rest position. When the object moves back again and compresses the spring, then now there is a compression in the spring, which once again is proportional to the compression, for the same reason.

Each particular instance of SHM needs its own explanation as to why the restoring force, and hence the acceleration, is proportional to the displacement from rest position, but this is the simplest version, and many others are not too different.

Exactly what the above post said, with a pendulum it is very similar. The resultant force acting on the pendulum is a component of weight which increases in magnitude as the angle the pendulum makes with the vertical increases - this means that the resultant force and hence acceleration are greatest when the pendulum is at its maximum displacement.