Question

Consider the situation shown in figure (5−E14). 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.

Figure

Answer 1

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
$$\begin{gathered} \Rightarrow \frac{Ma}{2}+Ma=Mg \\ \Rightarrow 3Ma=2Mg \\ \Rightarrow a=\frac{2g}{3} \end{gathered}$$

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


(c) Let T' = resultant of tensions

$$\begin{gathered} \therefore T\text{'}=\sqrt{T^2+T^2}=\sqrt{2}T \\ \therefore T\text{'}=\sqrt{2}T=\frac{\sqrt{2}Mg}{3} \\ \mathrm{Again},\tan \theta =\frac{T}{T}=1 \\ \Rightarrow \theta =45° \end{gathered}$$
So, the force exerted by the clamp on the pulley is $\frac{\sqrt{2}\mathrm{M}g}{3}$ at an angle of 45° with the horizontal.