Two small spheres of mass m connected by a string of length 2 $ℓ$ are kept on a smooth 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?

Question

TestHub · Accepted Answer

Let T be the tension in the string and θ the angle each half of the string makes with the horizontal surface.

Vertical balance of the midpoint where F is applied:

$F = 2 T sin θ$

Vertical balance of each sphere (N = normal reaction from the surface):

$N + T sin θ = m g$

$N = m g - T sin θ = m g - \frac{F}{2}$

For the spheres to just remain in contact with the surface, N ≥ 0:

$m g - \frac{F}{2} \geq 0 \Rightarrow F \leq 2 m g$

Hence the maximum value of F for which the spheres do not lose contact is 2mg.

Therefore option (A) is correct.

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