A solid hemisphere rests in equilibrium on a rough ground and against a smooth wall. The curved surface touches the wall and the ground. The angle of inclination of the circular base to the horizontal is 30° , The minimum coefficient of friction required between ground and hemisphere is μ . What is the value of 32 μ ?

Question

<img class="img-fluid question-image" alt="Question" src="https://cdn.tardigrade.in/q/nta/p-cj8zupn05thvi8xr.jpg" /> A solid hemisphere rests in equilibrium on a rough ground and against a smooth wall. The curved surface touches the wall and the ground. The angle of inclination of the circular base to the horizontal is 30° , The minimum coefficient of friction required between ground and hemisphere is μ . What is the value of 32 μ ? (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

First, make the free body diagram, of the solid hemisphere in equilibrium on rough ground and against the smooth wall, Solution

then the equation from above FBD are,

f = m g \times \frac{3 R}{8} cos 6 0^{o}

N_{2} = f

then normal reaction force,

N_{1} = m g

f = \frac{3 m g}{16}

f \leq μ N_{1}

μ \geq \frac{3}{16} \Rightarrow 32 μ = 32 \times \frac{3}{16} = 6

First, make the free body diagram, of the solid hemisphere in equilibrium on rough ground and against the smooth wall, then the equation from above FBD are, f=mg×83R​cos60o N2​=f then normal reaction force, N1​=mg f=163mg​ f≤μN1​ μ≥163​⇒32μ=32×163​=6