Question

Two blocks of masses M and m are arranged as shown in Fig. There is no friction between ground and block M.
The coefficient of static and Kinetic friction between M and m are ${\mu }_s$ and ${\mu }_k$ respectively.
a. Calculate the maximum possible value of F so that both the bodies move together.
b. Find the accelerations of the blocks if F is greater than that found in part (a).

Answer 1

If both blocks move together, i.e. no sliding between M and m, the friction between m and M will be static nature. Static friction force is self-adjusting force $0\le f\le {\mu }_{s}N$

Acceleration of system in this case, $a=\frac{F}{(M+m)}$ (i)

FBD of m and M:

Equation of motion of m:

$f=ma=m\frac{F}{(M+m)}$ (ii)

If there is no sliding between M and m, then $f\le f_{s-max}$

$\Rightarrow \frac{mF}{(M+m)}\le \mu \left(mg\right)$

$\Rightarrow F\le {\mu }_s(M+m)g$

b. If $F>{\mu }_s(M+m)g$, then there will be relative sliding between M and m starts.

The friction force reaches limiting value the friction force becomes $f_k-{\mu }_{k}N$

Free-body diagrams of m and M.

Equation of motion of $m:{\mu }_k\left(mg\right)=m{a}_1$

Acceleration of $m,a_1={\mu }_{k}g$

Equation of motion of $M:F-{\mu }_{k}mg=M{a}_2$

Acceleration of$M,a_2=\frac{F-{\mu }_{k}mg}{M}$