A 100 gm sphere D is at rest relative to drum ABC which rotates at a constant rate about its centre. Neglecting friction, find the velocity of the sphere if normal reaction exerted by slanting portion of drum is \frac{15 \sqrt{3}}{2} N

Engineering

Physics

Circular Motion in Vertical Plane

Horizontal Circular Motion

Centripetal and Centrifugal Force

Question

A 100 gm sphere D is at rest relative to drum ABC which rotates at a constant rate about its centre. Neglecting friction, find the velocity of the sphere if normal reaction exerted by slanting portion of drum is $\frac{15 \sqrt{3}}{2}$ N

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The sphere is at rest relative to the rotating drum, meaning it is in uniform circular motion. The net force towards the center provides the centripetal force. The weight (mg) and the normal force (N) from the slanting wall are the only forces acting. Resolving forces along the radial direction (towards center O) gives the centripetal force equation.

Given: m = 0.1 kg, N = (15√3)/2 N, θ = 60°.

Radial force balance: $N \cdot \cos (60 °) = \frac{m v^{2}}{r}$

From geometry, r = 0.3 m. Substituting values:

$\frac{15 \sqrt{3}}{2} \cdot \frac{1}{2} = \frac{0.1 \cdot v^{2}}{0.3}$

Solving: $v^{2} = \frac{15 \sqrt{3} \cdot 0.3}{2 \cdot 2 \cdot 0.1} = \frac{4.5 \sqrt{3}}{0.4} = 11.25 \sqrt{3}$

Final Answer: $v = 4.37 m / s$