A simple pendulum having a bob of mass m undergoes small oscillations with amplitude theta_0θ0. 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

t = -m (l omega^2+g cos theta)t=m(lω2+gcosθ)

Explanation:

For any thetaθ (bob angle with the vertical) we have

m vec alpha = vec T + vec Pmα=T+P

where

vec T = t(sintheta,costheta)T=t(sinθ,cosθ)
vec P = (0,-mg)P=(0,mg)
vec r = l (sin theta, cos theta)r=l(sinθ,cosθ)
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)