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)

1 Answer
Mar 19, 2018

Answer:

#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)#