Figure shows a small block A of mass m kept at the left end of a plank B of mass M = 2 m and length l. The system can slide on a horizontal road. The system is started towards right with the initial velocity v. The friction coefficients between the road and the plank is 1/2 and that between the plank and the block is 1/4. Find
(a) the time elapsed before the block separates from the plank.
(b) displacement of block and plank relative to ground till that moment.

Question

Accepted Answer

There will be relative motion between block and plank and plank and road. So at each surface limiting friction will act. The direction of friction forces at different surfaces are as shown in figure.

Here, [latex]f_1=\left(\frac{1}{4}\right)(m g)[/latex]

and [latex]f_2=\left(\frac{1}{2}\right)(m+2 m) g=\left(\frac{3}{2}\right) m g[/latex]

Retardation of A is [latex]a_1=\frac{f_1}{m}=\frac{g}{4}[/latex]

and retardation of B is [latex]a_2=\frac{f_2-f_1}{2 m}=\frac{5}{8} \mathrm{~g}[/latex]

Since, [latex]a_2>a_1[/latex]

Relative acceleration of A with respect to B is

[latex]a_r=a_2-a_1=\frac{3}{8} \mathrm{~g}[/latex]

Initial velocity of both A and B is v. So, there is no relative initial velocity. Hence,

(a) Applying [latex]s=\frac{1}{2} a t^2[/latex]

or [latex]l=\frac{1}{2} a_r t^2=\frac{3}{16} g t^2[/latex]

∴ [latex]t=4 \sqrt{\frac{l}{3 g}}[/latex]

(b) Displacement of block [latex]s_A=u_A t-\frac{1}{2} a_A t^2[/latex]

or [latex]s_A=4 v \sqrt{\frac{l}{3 g}}-\frac{1}{2} \cdot \frac{g}{4} \cdot\left(\frac{16 l}{3 g}\right)[/latex]. [latex]\left(a_A=a_1=\frac{g}{4}\right)[/latex]

or [latex]s_A=4 v \sqrt{\frac{l}{3 g}}-\frac{2}{3} l[/latex]

Displacement of plank [latex]s_B=u_B t-\frac{1}{2} a_B t^2[/latex]

or [latex]s_B=4 v \sqrt{\frac{l}{3 g}}-\frac{1}{2}\left(\frac{5}{8} g\right)\left(\frac{16 l}{3 g}\right)[/latex]. [latex]\left(a_B=a_2=\frac{5}{8} g\right)[/latex]

or [latex]s_B=4 v \sqrt{\frac{l}{3 g}}-\frac{5}{3} l[/latex]

Question 8.SE.24: Figure shows a small block A of mass m kept at the left end ...

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