Two block problem

Question

A plank having mass M is placed on smooth horizontal surface. Block of mass m is placed on it coefficient of friction between block and plank is ${\mathrm{\mu}}_{0}+{\mathrm{kx}}^{2}$, where k is constant and x is relative displacement between block and plank. A force F is applied on block where F = at, where a = 10; t is in second. Find ${\mathrm{t}}_{0}$ when relative motion will occur between block and plank

Moderate

Solution

Let at time ${\mathrm{t}}_{0}$ relative motion will occur

${\mathrm{\mu}}_{0}\mathrm{mg}=\mathrm{Ma}\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}......\left(1\right)$

$10\text{\hspace{0.17em}}{\mathrm{t}}_{0}-{\mathrm{\mu}}_{0}\mathrm{mg}=\mathrm{ma}\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}.......\left(2\right)$

From (1) and (2) ;

${\mathrm{t}}_{0}={\mathrm{\mu}}_{0}\mathrm{m}+\frac{{\mathrm{\mu}}_{0}{\mathrm{m}}^{2}}{\mathrm{M}}$

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