The friction coefficient between the two blocks shown in the following figure is μ 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.

#### Solution

From the free body diagrams of the two blocks, we have

R_{1} = mg ...(i)

F = μR_{1}+T ...(ii)

T − μR_{1} = 0 ..(iii)

From equations (i) and (ii), we have

F − μmg = T ...(ii)

From equations (i) and (iii), we have

T = μmg

Putting T = μmg in equation (ii), we have

F = μmg + μmg = 2μmg

(b) From the free body diagram of upper block, we have

2F − T − μmg = ma ....(i)

From the free body diagram of lower block, we have

T = Ma + μmg

Putting the value of T in (i), we get

2F − Ma − μmg − μmg = ma

Putting F = 2μmg, we get

2(2μmg) − 2μmg = a(M + m)

⇒ 4μmg − 2μmg = a(M + m)

`=> a=(2mu "mg")/("M"+"m")`in opposite directions.