Three point particles $P, Q, R$ move in circle of radius $‘r’$ with different but constant speeds. They start moving at $t = 0$ from their initial positions as shown in the figure. The angular velocities (in rad/ sec) of $P, Q$ and $R$ are $5\pi , 2\pi$ & $3\pi$ respectively, in the same sense. The time interval after which they are at same angular position.

A car changes speed from $18\,km/h$ to $36\,km/h$ in $5\,s$. The diameter of its wheel is $0.8\,m$ . The angular acceleration of the wheel is ........ $rad/s^2$

A tube of length $L$ is filled completely with an incompressible liquid of mass $M$ and closed at both the ends. The tube is then rotated in a horizontal plane about one of its ends with a uniform angular velocity $\omega $. The force exerted by the liquid at the other end is

Two cars of masses $m_1$ and $m_2$ are moving in circles of radii $r_1$ and $r_2$, respectively. Their speeds are such that they make complete circles in the same time $t$. The ratio of their centripetal acceleration is

A horizontal curve on $a$ racing track is banked at a $45^o $ angle. When a vehicle goes around this curve at the curve’s safe speed (no friction needed to stay on the track), what is its centripetal acceleration?

A boy ties a stone of mass $100 \,g$ to the end of a $2$ $m$ long string and whirls it around in a horizontal plane. The string can withstand the maximum tension of $80 \,N$. If the maximum speed with which the stone can revolve is $\frac{ K }{\pi} rev$. / $min$. The value of $K$ is (Assume the string is massless and unstretchable)