Consider the situation shown in the following 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.

#### Solution

Let the acceleration of mass M be a.

So, the acceleration of mass 2M will be \[\frac{a}{2}\]

(a) 2M(a/2) − 2T = 0

⇒ Ma = 2T

T + Ma − Mg = 0

\[\Rightarrow \frac{Ma}{2} + Ma = Mg \]

\[ \Rightarrow 3Ma = 2Mg\]

\[ \Rightarrow a = \frac{2g}{3}\]

(b) Tension,

\[T = \frac{Ma}{2} = \frac{M}{2} \times \frac{2g}{3} = \frac{Mg}{3}\]

(c) Let T' = resultant of tensions

\[\therefore T' = \sqrt{T^2 + T^2} = \sqrt{2}T\]

\[ \therefore T' = \sqrt{2}T = \frac{\sqrt{2}Mg}{3}\]

\[\text{Again, }\tan\theta = \frac{T}{T} = 1\]

\[ \Rightarrow \theta = 45^\circ\]

So, the force exerted by the clamp on the pulley is `(sqrt2"Mg")/3` at an angle of 45° with the horizontal.