Question

Consider the situation shown in the figure. Both the pulleys and the string are light and all the surfaces are smooth.
(a) Find the acceleration of the mass $M$ .
(b) Find the tension in the string.
(c) Calculate the force exerted by the clamp on the pulley $A$ in the figure.

Answer 1

(a)

The acceleration is given as,

$T+Ma-Mg=0$

$\frac{Ma}{2}+Ma-Mg=0$

$a=\frac{2g}{3}$

(b)

The tension is given as,

$T=\frac{Ma}{2}$

$T=\frac{M\times \frac{2g}{3}}{2}$

$T=\frac{Mg}{3}$

(c)

Let the resultants of the tension is $R$.

It can be written as,

$R^{\prime }=\sqrt{T^2+T^2}$

$R^{\prime }=\frac{\sqrt{2}Mg}{3}$

The angle is given as,

$\tan \theta =\frac{T}{T}$

$\theta ={45}^{\circ }$

Thus, the force exerted by the clamp on the pulley is $\frac{\sqrt{2}Mg}{3}$ at an angle of ${45}^{\circ }$ with horizontal.