Question

In the following figure shows a small block of mass m kept at the left end of a larger block of mass M and length l. The system can slide on a horizontal road. The system is started towards right with an initial velocity v. The friction coefficient between the road and the bigger block is μ and that between the block is μ/2. Find the time elapsed before the smaller blocks separates from the bigger block.

Answer 1

Let a₁ and a₂ be the accelerations of masses m and M, respectively.
Also, a₁ > a₂ so that mass m moves on mass M.

Let after time t, mass m is separated from mass M.

Using the equation of motion
During this time, mass m covers `ct+1/2a_1t^2` and `s_m=vt+1/2a_2t^2`.

For mass m to separate from mass M, we have: `vt+1/2a_1t^2 = vt+1/2a_2t^2+1`                ........(ii)

From the free body diagram, we have:
`ma_1+mu/2"R"=0`

`=>ma_1=-(mu/2)"mg"=(mu/2)mxx10`a₁ = - 5µ

Again,
`Ma_2+mu("M""m")g-(mu/2)mg=0`

⇒ 2Ma₂ + 2μ (M + m)g − μmg = 0
⇒ 2Ma₂ = μmg − 2μmg − 2μmg

`=> a_2=(-mumg-2muMg)/(2M)`

Substituting the values of a₁ and a₂ in equation (i), we get: 
`t = sqrt(4Ml)/((M+m)mug)`