Board-(a) is lying on top of board-(b). Together, they form a single unit and move with some velocity v on a smooth horizontal surface. They collide with another identical board-(c). There is no frict

Question

Board-(a) is lying on top of board-(b). Together, they form a single unit and move with some velocity v on a smooth horizontal surface. They collide with another identical board-(c). There is no friction between boards-(a) and (b), but there is friction between boards-(a) and (c). Upon collision, boards-(b) and (c) are firmly coupled and do not move relative to each other. The length of each board is L. Board -(a) stops moving relative to board-(b) and (c) at the moment it is entirely located on top of board-(c). Find the displacement $Δ x$ of board (a) from the moment of collision until the relative motion of the boards stops. If the value of $Δ x$ is $\frac{α L}{21}$ , the value of $α$ is. Take : $π = \frac{22}{7}$ . (all board are identical.)

TestHub · Accepted Answer

When (b) hits (c), the two boards couple:

$m v = 2 mV \Rightarrow V = \frac{v}{2}$

(a) keeps its initial speed v. Friction grows with its overlap x on (c):

$F = μ m g \frac{x}{L}$

Reduced-mass energy balance for relative motion:

$\frac{1}{2} \cdot \frac{2 m}{3} (\frac{v}{2})^{2} = \int_{0}^{L} μ m g \frac{x}{L} d x$

$= \frac{μ m gL}{2} \Rightarrow μg = \frac{v ^{2}}{6 L}$

The relative coordinate has angular frequency

$ω = \frac{3 μg}{2 L} = \frac{v}{2 L}$

It stops after a quarter cycle:

$ω t = \frac{π}{2}$

The centre of mass moves at 2v/3. At stopping, (a) has advanced L relative to (b,c).

$Δ x = \frac{2 v}{3} \cdot \frac{π L}{v} + \frac{2 L}{3} = \frac{2 L ( π + 1 )}{3}$

$= \frac{2 L}{3} (\frac{22}{7} + 1) = \frac{58 L}{21}$ .

Answer: $α = 58$ .

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