Block of mass rests on a rough ramp that can have its angle adjusted

PHYSICS · Multi-part · CBSE · Physics · Laws of Motion · Laws of Motion

Problem

Block

A

of mass

m_A

rests on a rough ramp that can have its angle

\theta

adjusted. A second block

B

of mass

m_B

is gently placed directly on top of block

A

so that it presses vertically downward but does not slide relative to

A

. The contact surfaces between

A

and the ramp have coefficient of static friction

\mu_s

. Friction between the two blocks is negligible. The ramp angle is slowly increased until block

A

is just about to begin sliding down the ramp. Answer the following questions, justifying each algebraic step. (a) Derive an expression for the normal force exerted by the ramp on block

A

in terms of

m_A

m_B

g

, and

\theta

. (b) Using the result from part (a), obtain a symbolic expression for the critical angle

\theta_c

at which block

A

starts to slide. Your final answer should be in terms of

\mu_s

and the ratio

\dfrac{m_A+m_B}{m_A}

. (c) Evaluate

\theta_c

numerically for

m_A = 3.0\,\text{kg}

m_B = 1.5\,\text{kg}

, and

\mu_s = 0.40

. Express your answer to the nearest degree.

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