Two racing 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 each makes a complete circle in the same time $t$. The ratio of the angular speeds of the first to the second car is
$1 : 1$
$m_1 : m_2$
$r_1 : r_2$
$m_1m_2 : r_1r_2$
In free space, a shell moving with velocity $60\,m/s$ along the positive $x$ -axis. When it passes through origin, it explodes into two peices of mass ratio $1 : 2.$ After the explosion, the velocity of the centre of mass is ........ $m/s$.
A ring of radius $4a$ is rigidly fixed in vertical position on a table. A small disc of mass $m$ and radius $a$ is released as shown in the fig. When the disc rolls down, without slipping, to the lowest point of the ring, then its speed will be
A child is standing at one end of a long trolley moving with a speed $v$ on a smooth horizontal floor. If the child starts running towards the other end of the trolley with a speed $u$, the centre of mass of the system (trolley + child) will move with a speed.
A particle originally at rest at the highest point of $a$ smooth vertical circle is slightly displaced. It will leave the circle at $a$ vertical distance $h$ below the highest point, such that
A uniform rod of mass $m$ and length $l$ rotates in a horizontal plane with an angular velocity $\omega $ about a vertical axis passing through one end. The tension in the rod at a distance $x$ from the axis is