Question

A string of negligible mass passing over a clamped pulley of mass m supports a body of mass M as shown in figure. The force exerted by the clamp on the pulley is.

• A. $g \sqrt{(M + m)^2 + M^2 } $
• B. $\sqrt{(M + m)^2 + m^2 } g$
• C. $\sqrt{2}$ Mg
• D. $\sqrt{2}$ mg

Answer 1

Answer: (A)

A free body diagram of a mass pully system is shown below

Now, tension in the string
$T - Mg = 0$
$\Rightarrow T = Mg$
Force in vertical direction,
$F_1 = T + mg $
$\Rightarrow F_1 = Mg + mg = ( M+ m) g$
Similarly, force in horizontal direction
$F_2 = T = Mg $
Now, the resultant force
$F_{res} = \sqrt{F_{1}^{2} + F_{2}^{2} + 2F_{1}F_{2} cos \,90^{\circ}} $
$ \Rightarrow F_{res} = \sqrt{F_{1}^{2} +F_{2}^{2}} = \sqrt{\left(M+m\right)^{2} g^{2} +M^{2}g^{2}}$
$ \Rightarrow F_{res} =g \sqrt{\left(M+m\right)^{2} +M^{2} }$
Hence, the resultant force is $g \sqrt{\left(M+m\right)^{2} +M^{2} }$