First slide

Equilibrium of a particle

Question

Two small spheres each of mass m connected by a string of length 2l are kept on a smooth horizontal surface. A vertical force F is applied at the middle of the string. What is maximum value of F for which the spheres do not lose contact with the surface?

Moderate

A

$2 m g$
B

$m g$
C

$\frac{3 m g}{2}$
D

$4 m g$

Solution

$F = 2 T \sin θ or T \sin θ = \frac{F}{2} . . . . . . (i)$
$\begin{array}{l} T \sin θ + N = m g . . . . . . . (i i) \\ \Rightarrow N = m g - T \sin θ \end{array}$
For no loss of contact:
$\begin{array}{l} N > 0 \\ \Rightarrow m g - T \sin θ > 0 \\ \Rightarrow m g - \frac{F}{2} > 0 \\ \Rightarrow F < 2 m g \\ So F_{max} = 2 m g \end{array}$

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