A particle moving with uniform speed in a circular path maintains:

The angular speed of seconds needle in a mechanical watch is

The hour hand of a clock is $6\,cm$ long. The magnitude of the displacement of the tip of hour between $1:00\,PM$ to $5:00\,PM$ is

Two particles having mass $M$ and $m$ are moving in a circular path having radius $R$ and $r$. If their time period are same then the ratio of angular velocity will be

A student skates up a ramp that makes an angle $30^{\circ}$ with the horizontal. $He /$ she starts (as shown in the figure) at the bottom of the ramp with speed $v_0$ and wants to turn around over a semicircular path xyz of radius $R$ during which he/she reaches a maximum height $h$ (at point y) from the ground as shown in the figure. Assume that the energy loss is negligible and the force required for this turn at the highest point is provided by his/her weight only. Then ( $g$ is the acceleration due to gravity)

$(A)$ $v_0^2-2 g h=\frac{1}{2} g R$

$(B)$ $v_0^2-2 g h=\frac{\sqrt{3}}{2} g R$

$(C)$ the centripetal force required at points $x$ and $z$ is zero

$(D)$ the centripetal force required is maximum at points $x$ and $z$

A sphere of mass $m$ is tied to end of a string of length $l$ and rotated through the other end along a horizontal circular path with speed $v$. The work done in full horizontal circle is