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In figure, a body A of mass m slides on a rough plane inclined at angle $θ_{1}$ to the horizontal and μ is the coefficient of friction between A and the plane. A is connected by a light string passing over a frictionless pulley to another body B, also of mass m, sliding on a frictionless plane inclined at an angle $θ_{2}$ to the horizontal. Which of the following statements are true?

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A will never move up the plane

A will just start moving up the plane when $μ = \frac{\sin θ_{2} - \sin θ_{1}}{\cos θ_{1}}$

For A to move up the plane, $θ_{2}$ must always be greater than $θ_{1}$

B will always slide down with constant speed

answer is B, C.

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Detailed Solution

Let A moves up the plane frictional force on A will be downward as shown.

When A just starts moving up

$m g \sin θ_{1} + f = m g \sin θ_{2} \Rightarrow m g \sin θ_{1} + μ m g \cos θ_{1} = m g \sin θ_{2} \Rightarrow μ = \frac{\sin θ_{2} - \sin θ_{1}}{\cos θ_{1}}$

When A moves upwards

$f = m g \sin θ_{2} - m g \sin θ_{1} > 0 \Rightarrow \sin θ_{2} > \sin θ_{1} \Rightarrow θ_{2} > θ_{1}$

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