Answer to Question #190830 in Mechanics | Relativity for yelivi

Question

Show that the maximum tension in the string of a simple pendulum, when the

amplitude θ is small, is mg(1+(θ)^2). At what position of the pendulum is the tension a

maximum?

Answer 1

Tension of the string when the string reaches the mid point

$T=T_m$

$T_m=mg+\dfrac{mv^2}{l}$

Applying energy balance

$\dfrac{1}{2}mv^2=mgl(1-cos\theta)$

$v^2=2gl(1-cos\theta)$

$v^2=2gl\Big(2sin^2\dfrac{\theta}{2}\Big)$

Substituting the value of $v^2$ in above equation

$T_m=mg+2mgl\dfrac{\Big(2sin^2\dfrac{\theta}{2}\Big)}{l}$

$T_m=mg+4mg\Big(sin^2\dfrac{\theta}{2}\Big)$

$T_m=mg\Big(1+4sin^2\dfrac{\theta}{2}\Big)$

for small $\theta, \space sin\theta\approx\theta$

$T_m=mg\Big(1+4\Big(\dfrac{\theta}{2}\Big)^2\Big)$

$T_m=mg(1+\theta^2)$

The tension of the string is maximum at extreme positions