Question

Consider the situation in figure. Both the pulleys and the string are light and all the surfaces are frictionless. (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

$M_a-2T=0.T+Ma-Mg=0$

$\Rightarrow$ Ma =2T

$\Rightarrow \frac{Ma}{2}+Ma=Mg$ (because T $=\frac{Ma}{2}$)

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

$\Rightarrow$ 3Ma = 2Mg

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

(a) Acceleration of mass,

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

(b) Tension, T = $\frac{Ma}{2}=\frac{M}{2}\times \frac{2g}{3}=\frac{Mg}{3}$

(c) Let, R' = resultant of tensions
= force exerted by the
clamp on the pulley

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

$\therefore R=\sqrt{2}T=\frac{\sqrt{2}Mg}{3}$

Again, $\tan \theta =\frac{T}{T}=1$

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

So, it is $\frac{\sqrt{2}Mg}{3}$ at an angle of ${45}^{\circ }$ with horizontal.