Question

The friction coefficient between the two blocks shown in figure (6-E9) is $\mu$ but the floor is smooth.
(a) What maximum horizontal force F can be applied without disturbing the equilibrium of the system?
(b) Suppose the horizontal force applied is double of that found in Part (a). Find the accelerations of the two masses.

Answer 1

Both upper block and lower block will have acceleration 2 m/s^2.

$R_1=mg,f=\mu R_1-T=0\Rightarrow T-\mu R_1=0$
and $F-\mu mg=T\Rightarrow T=\mu mg$
Putting $T=\mu mg,F=\mu mg+\mu mg=2\mu mg$
(b) $2F=T-\mu mg-ma=0T-ma-\mu mg=0\Rightarrow T=ma+\mu mg$
Putting value of T in (i), we have
$2F-Ma-\mu mg-\mu mg-ma=0$
Putting $F=2\mu mg$
$2\left(2\mu mg\right)-2\mu mg=a(M+m)$
$\Rightarrow 4\mu mg-2\mu mg=a(M+m)$
$\Rightarrow a=\frac{2\mu mg}{M+m}$
Both blocks move with this acceleration 'a' in opposite directions.