Question

A simple pendulum has a bob of mass m and swings with an angular amplitude $\varphi$. The tension in the thread is T. At a certain time, the string makes an angle $\theta$ with the vertical ($\theta \le \varphi$)

• A. $T=mg\cos \theta$, for all values of $\theta$
• B. $T=mg\cos \theta$, only for $\theta =\varphi$
• C. $T=mg$, for $\theta ={\cos }^{-1}\left[\frac{1}{3}\right(2cos\varphi +1\left)\right]$
• D. T will be larger for smaller values of $\theta$

Answer 1

Answer: (B), (C), (D)

The correct options are
B $T=mg$, for $\theta ={\cos }^{-1}\left[\frac{1}{3}\right(2cos\varphi +1\left)\right]$
C T will be larger for smaller values of $\theta$
D $T=mg\cos \theta$, only for $\theta =\varphi$

At any point of time the forces on the bob are Tension , gravitational

and the centripetal force

We have

$mg(lcos\theta -lcos\varphi )=1/2m{v}^2\frac{m{v}^2}{l}=2mg(cos\theta -cos\varphi )AlsoT-mgcos\theta =\frac{m{v}^2}{l}=2mg(cos\theta -cos\varphi )orT=mg(3cos\theta -2cos\varphi )$

For $\theta$ = ${\cos }^{-1}\frac{1}{3}(2cos\varphi +1)$

$T=mg$