Question

a pendulum consisting of a small sphere of mass M suspended by an inextensible and massless string of length l is made to swing in a vertical plane. if the breaking tension of the string is 2Mg , then the maximum angular amplitude of the displacement from the vertical curve is?

Answer 1

Answer: The maximum angular amplitude will be $\theta = 60^{\circ}$.

Explanation:

Given that,

Mass of sphere = M

Length of string = l

Tension T = 2 Mg

We know that,

Tension at the lowest point is

$T = \dfrac{mv^2}{r}+mg$

$2Mg=\dfrac{Mv^2}{l}+Mg$

$Mg=\dfrac{Mv^2}{l}$

$v = \sqrt{gl}$

Now, equate the potential energy and kinetic energy

Potential energy at the top point = kinetic energy at the lowest point

$mgh=\dfrac{1}{2}mv^2$

$mgl(1-\cos\theta)=\dfrac{1}{2}m(gl)$

$1-\cos\theta = \dfrac{1}{2}$

$\cos\theta = \dfrac{1}{2}$

$\theta = 60^{\circ}$

Hence, The maximum angular amplitude will be $\theta = 60^{\circ}$.