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.
