Question

A bead A can slide freely along a smooth rod bent in the form of a half circle of Radius R. The system is set in rotation with a constant angular velocity ω about a vertical axis OO'. Find the angle θ corresponding to steady position of the bead.

• A. cos^-1 $\left( \frac { \mathrm { R } \omega ^ { 2 } } { \mathrm {~g} } \right)$
• B. cos^-1 $\left( \frac { g } { R \omega ^ { 2 } } \right)$
• C. sin^-1 $\left( \frac { g } { R \omega ^ { 2 } } \right)$
• D. cos^-1 $\left( \frac { \mathrm { R } \omega ^ { 2 } } { \mathrm {~g} } \right)$

Answer 1

Answer: (B)

cos^-1 $\left( \frac { g } { R \omega ^ { 2 } } \right)$

Answer 2

force acting along the tangent to the radius r = R sin θ.

mg sin θ - mr ω² cos θ = 0

or mg sin $\left[ 1 - \frac { \mathrm { r } \omega ^ { 2 } } { \mathrm {~g} } \cos \theta \right]$ = 0 or θ = cos^-1 $\left( \frac { g } { R \omega ^ { 2 } } \right)$