A car is moving at a speed of $40 \,m / s$ on a circular track of radius $400 \,m$. This speed is increasing at the rate of $3 \,m / s ^2$. The acceleration of car is ....... $m / s ^2$

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 $z$

The centripetal acceleration is given by

A body is revolving with a uniform speed $v$ in a circle of radius $r$. The tangential acceleration is

A ball of mass $0.5 \mathrm{~kg}$ is attached to a string of length $50 \mathrm{~cm}$. The ball is rotated on a horizontal circular path about its vertical axis. The maximum tension that the string can bear is $400 \mathrm{~N}$. The maximum possible value of angular velocity of the ball in rad/s is,:

Two bodies of mass $10 \,kg$ and $5 \,kg$ moving in concentric orbits of radii $R$ and $r$ such that their periods are the same. Then the ratio between their centripetal acceleration is