Question

A simple pendulum having a bob of mass m undergoes small oscillations with amplitude `theta_0`. Find the tension in the string as a function of the angle made by the string with the vertical?

(Try not to use the energy conservation method)

Answer 1

`t = -m (l omega^2+g cos theta)`

Explanation:

For any `theta` (bob angle with the vertical) we have

`m vec alpha = vec T + vec P`

where

`vec T = t(sintheta,costheta)`
`vec P = (0,-mg)`
`vec r = l (sin theta, cos theta)`
#vec alpha = ddot(vec r) = l(-Sin theta omega^2 + Cos theta dotomega, - (Cos theta omega^2 + Sin theta dotomega))#

with `omega = dot theta`

or

`{(l m ( Cos theta dotomega-Sin theta omega^2) = t Sin theta),(-l m ( Sin theta dotomega+Cos theta omega^2) = m g + t Cos theta):}`

now solving for `ddot theta, t` we obtain

`ddot theta = -(m g)/l sin theta, t = -m (l omega^2+g cos theta)`

hence

`t = -m (l omega^2+g cos theta)`