Question

A sphere of mass m and radius R rests on sufficiently rough inclined plane in equilibrium as shown in the figure. Find the tension in the string.

Answer 1

Given: a sphere of mass m and radius R rests on sufficiently rough inclined plane in equilibrium as shown in the figure.

To find the tension in the string

Solution:

The above figure shows the free body diagram

Now apply $\sum \overrightarrow{T}=0=0$ about an axis through the center of the sphere.

And $\sum F_x=0$ to the sphere

Substitute for f:

$fR-TR=0⟹T=f⟹f+T\cos \theta -mg\sin \theta =0⟹T=\frac{mg\sin \theta }{1+\cos \theta }⟹T=\frac{m\times 10\times \sin \left(30\right)}{1+\cos \left(30\right)}⟹T=\frac{5m}{\frac{2+\sqrt{3}}{2}}⟹T=\frac{10m}{2+\sqrt{3}}$

is the tension in the string.