Question

Let us consider the next case in the previous illustration when there is no friction between ground and M. The coefficients of static and kinetic friction are ${\mu }_s$ and ${\mu }_k$, respectively, and force F acts on upper block as shown in Fig.
a. What is the maximum possible value of F so that the system moves together?
b. If there is relative sliding between M and m, then calculate acceleration of M and m.

Answer 1

Let the system move together, then $a=\frac{F}{(M=m)}$.
FBD of M: f=ma = $M(\frac{F}{M+m)}$ (i)
As there is no sliding between M and m,
$M(\frac{F}{M+m}\le {\mu }_{s}mg$
$\Rightarrow F\le {\mu }_s(M+m)g$
If $F>{\mu }_s\frac{m}{M}(M+m)g$, the relative sliding between the blocks starts. The friction forec between force between the blocks will be kinetic nature and both blocks will move with different acclerations.
Acceleration of m, $a_1=\frac{F-{\mu }_{k}mg}{m}$
Acceleration of M, $a_2=\frac{{\mu }_{k}mg}{M}$