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Question

Figure (6−E12) 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.

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Solution

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 vt+12a1t2 and sm=vt+12a2t2.

For mass m to separate from mass M, we have:
vt+12a1t2=vt+12a2t2+1 ....ii




From the free body diagram, we have:

ma1+μ2R=0ma1=-μ2mg=μ2m×10a1=-5μ

Again,
Ma2+μM+mg-μ2mg=0
⇒ 2Ma2 + 2μ (M + m)g − μmg = 0
⇒ 2Ma2 = μmg − 2μmg − 2μmg
a2=-μmg-2μMg2M

Substituting the values of a1 and a2 in equation (i), we get:
t=4MlM+mμg

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