Simple Harmonic Motion

If a body moves in such a way that its acceleration is directed towards a fixed point in its path and directly proportional to the distance from that point, the movement of the object is said to be simple harmonic

E.g.

When a simple pendulum swings to and fro, the acceleration of its bob is directed towards the centre point of its motion and is proportional to the distance from that point. Therefore, the motion of a simple pendulum is SHM.

As you can see, when the weight of the pendulum bob is resolved, the tension of the string, T, and the mg cos x cancel each other out, leaving mg sin x as the net force, as shown above. This force is responsible for bringing the bob down in a curved path.
Using F = ma for the bob,
mg sin x = ma, where a is the acceleration of the bob.
If the pendulum swings through a small angle and is measured in radians, sin x is almost equal to x.
mg. x = m a
gx = a
g d/l = a ( x = d / l radians)
a = (g/l) d
a = k d
a α d
The acceleration of the bob is directly proportional to the distance from the centre point. Therefore, the motion of a simple pendulum is simple harmonic.
k = ω² where ω is the angular speed.
a = ω² d
ω² = g/l
ω = √g/l
If the time period is T,
T = 2π/ω

T = 2π/ √g/l

So, the time period of a simple pendulum depends only on its length; it does not depend on the mass of the bob. The formula only works for the oscillations through small angles, as it was something we assumed in the process of proof.

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Please work out the following questions to complement what you have just learnt.

1) The spring constant of a spring - the force per unit extension - is k. It is attached to a ceiling and the free end is connected to a weight. The weight is pulled down and released. Show its motion is SHM and hence find an expression for the time period.
2) Two springs of spring constants p and q are attached to a ceiling. A mass is connected at the free ends that are at the same level. When the mass is pulled down and released, its motion becomes SHM. Find its period.
3) If the two strings in Question 2 are connected in series, find an expression for the time period.
4) A ball bearing is released from rest on the surface of a concave mirror. The diameter of the mirror is small. Show the motion of the ball bearing is SHM and hence derive an expression for its time period.
5) An object is placed on a smooth table. It is attached to two springs of spring constant p and q, parallel to each other. The other ends of springs are connected to a vertical wall. Show that, the object performs SHM when pulled away and released. Hence find an expression for the time period.