Question

A horizontal platform vibrates with simple harmonic motion in the horizontal direction with a period of 2 second .A body of mass 0.5 kg is placed on the platform is 0.3 . what is the maximum frictional force on the body when the platform is oscillating with an amplitude 0.2m ?

Answer 1

In the given problem the plat form is oscillating and the body on it also oscillates with platform. For this Frictional force is necessary.

When a body of mass 'm' is oscillating with angular velocity 'ω' and amplitude A, force required by the  body is F = $m\omega^2A$.

Angular frequency 'ω' is related to time period 'T' as 'ω' = $\frac{2\pi}{T}$.
so ω= $\frac{2\pi}{2} = \pi$ rads/s.

Therefore,  force required by the body  F = $0.5\times (\pi)^2\times\:0.2 = 0.98\: N$.

If coefficient of static friction, μ = 0.3 ( Not clearly given in the problem, so made a guess!!),
Then the maximum friction is μmg = [tex]0.3\times 0.5\times 9.8 = 1.47 N. [/tex].
This is larger than the required force. Therefore the body will definitely oscillate with the platform without slipping.
The frictional force on the body is equal to the force required for oscillation and is equal to 0.98 N.