Question

A wire is in the shape of a parabola $y=5x^2$.It is being rotated about y-axis with an angular velocity $\omega$ . A small Bead of mass (m=$1kg$) is at point P and is in equilibrium with respect to the fram. Then the angular velocity $\omega$ is $(g=10m/s^2)$

• A. $\omega =5rad/sec$
• B. $\omega =10rad/sec$
• C. $\omega =20rad/sec$
• D. $\omega =15rad/sec$

Answer 1

Answer: (B)

$\omega =10rad/sec$

Given: $m_1=1kg$

equation of curve $y=5x^2$

Slope of tangent at $P(1,5)=y^{\prime }$

$y_{(1,5)}^1=10x$

$y^{\prime }=10$

$\tan \theta =10$

Now from free body diagram.

For the condition of equilibrium along the tangent is

$\begin{array}{c}m{\omega }^{2}r\cos \theta =\mathrm{mgsin}\theta \\ \omega =\sqrt{\frac{g\tan \theta }{r}}\end{array}$

$[$ B) $\omega =10$ rad $/\sec$