(a) Define simple harmonic motion and derive, an expression for the period of simple harmonic motion by reference circle method.
(b) Derive an expression for the period of oscillations of a simple pendulum.

Answer

Simple pendulum, which consists of a particle of mass `m` (called the bob of the pendulum) suspended from one end of an unstretchable, massless string of length `L` fixed at the other end as shown in Fig. (a). The bob is free to swing to and fro in the plane of the page, to the left and right of a vertical line through the pivot point. The force acting on the bob are the force,`T`, tension in the string and the gravitational force `F_(g) (= m g)`, as shown in Fig. (b).
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Torque `tau` about the support is entirely provided by the tangental component of force
`tau = -L (mg din theta)`
This is a restoring torque that tendsd to reduce angular displcement, hence the negative sing. By Newton's law of rotational motion.
`tau = I alpha`
Where `I` is the moment of intertial of the system about the support and `alpha` is the angular acceleration. Thus,
`Ialpha = - mg sin theta L`
or `alpha = -(mgL)/(I) sintheta`
Now if `theta` is small, `sintheta` cab be approximated by `theta` and Eq. can then be written as,
`alpha = -(mgL)/(I)theta.....(1)`
For small `theta` the motion of the bob is angular simple harmonic motion
`alpha = -omega^(2)theta ....(2)`
From eq. `(1)` and `(2) omega = sqrt((mgL)/(I))`
and `T = 2pisqrt((I)/(mgL.))`