$A$ particle inside the rough surface of $a$ rotating cone about its axis is at rest relative to it at $a$ height of $1m$ above its vertex. Friction coefficient is $\mu = 0.5$, if half angle of cone is $45^o$, the maximum angular velocity of revolution of cone can be :

$A$ particle is moving in a circle :

A cyclist riding the bicycle at a speed of $14 \sqrt{3} \,m / s$ takes a turn around a circular road of radius $20 \sqrt{3} \,m$ without skidding. What is his inclination to the vertical?

Obtain the formula for the maximum safe speed $(v_{max})$ of a vehicle on a level curved road.

A ball of mass $0.1\, Kg$. is whirled in a horizontal circle of radius $1\, m$. by means of a string at an initial speed of $10\, R.P.M.$ Keeping the radius constant, the tension in the string is reduced to one quarter of its initial value. The new speed is ....... $r.p.m.$

A body of mass $'m '$ is tied to one end of a spring and whirled round in a horizontal plane with a constant angular velocity. The elongation in the spring is $1\, cm$. If the angular velocity is doubled, the elongation in the spring is $5\, cm$. The original length of the spring is .........  $cm$