A tube of length $L$ is filled completely with an incompressible liquid of mass $M$ and closed at both ends. The tube is then rotated in a horizontal plane about one of its end with a uniform angular velocity $\omega $ . Then the force exerted by the liquid at this other end is
$M{\omega ^2}L$
$\frac{1}{2}M{\omega ^2}L$
$\frac{1}{4}M{\omega ^2}L$
$2M{\omega ^2}L$
Particles of masses $m, 2m, 3m, ...... nm$ $grams$ are placed on the same line at distances $l, 2l, 3l,...., nl\, cm$ from a fixed point. The distance of the centre of mass of the particles from the fixed point (in centimetres) is
Two particles whose masses are $10\,kg$ and $30\,kg$ and their position vectors are $\hat i +\hat j+ \hat k$ and $-\hat i -\hat j -\hat k$ respectively would have the centre of mass at
A wheel of radius $r$ rolls without slipping with a speed $v$ on a horizontal road. When it is at a point $A$ on the road, a small jump of mud separates from the wheel at its highest point $B$ and drops at point $C$ on the road. The distance $AC$ will be
A hoop of radius $r$ and mass $m$ rotating with an angular velocity ${\omega _0}$ is placed on a rough horizontal surface. The initial velocity of the centre of the hoop is zero. What will be the velocity of the centre of the hoop when it ceases to slip?
A uniform bar of length $'6l'$ and mass $'8m'$ lies on a smooth horizontal table. Two point masses $m$ and $2m$ moving in the same horizontal plane with speed $2v$ and $v$ respectively, strike the bar (as shown in the fig.) and stick to the bar after collision. Total energy (about the centre of mass, $c$ ) will be