Question

A trolley of mass M oscillates with some time period about its mean position when it is connected with an ideal spring of stiffness constant K.

If an unknown mass m is gently placed at the top of trolley, its time period is found to be T.Calculate the coefficient of static friction so that the upper mass does not slip even for sizeable amplitude A.

• A. $\frac{KA}{mg}$
• B. $\frac{KA}{(M+m)g}$
• C. $\frac{KAm}{mg}$
• D. None of these

Answer 1

The correct option is B

$\frac{KA}{(M+m)g}$

As there is no relative slipping for small oscillations with time period T.

$T=2\pi \sqrt{\frac{M+m}{k}}$

$\omega =\sqrt{\frac{K}{(M+m)}}$

Now, for large amplitude A (limiting case) $\Rightarrow$ maximum acceleration = ${\omega }^{2}A$

Which will be provided by friction $\Rightarrow \mu g={\omega }^{2}A$

$\Rightarrow \mu =\frac{{\omega }^{2}A}{g}=\frac{KA}{(M+m)g}$