Exercises

A conical pendulum consists of a simple pendulum moving in a horizontal circle as shown. $C$ is the pivot, $O$ the centre of the circle in which the pendulum bob moves and $\omega$ the constant angular velocity of the bob. If $\overrightarrow{L}$ is the angular momentum about point $C$, then

A plank $P$ is placed on a solid cylinder $S$, which rolls on a horizontal surface. The two are of equal mass. There is no slipping at any of the surfaces in contact. The ratio of the kinetic energy of $P$ to the kinetic energy of $S$ is

Two wheels $A$ and $B$ are released from rest from points $X$ and $Y$ respectively on an inclined plane as shown in the figure. Which of the following statement(s) is/are incorrect?

A solid sphere is given an angular velocity $\omega$ and kept on a rough fixed incline plane. Then choose the correct statement.

A wheel $A$ starts rolling up a rough inclined plane and another identical wheel $B$ starts rolling up a smooth plane having same inclination with the horizontal. If initial velocity of both the wheels is same then

Consider a fixed wheel of radius $R$ . A small wheel (in the form, of a uniform solid disc) $r$of radius  is performing pure rolling on periphery of bigger wheel. centers of bigger wheel and smaller wheel are joined by rigid rod such that smaller wheel can rotate freely with respect to its centre. Rod joining the (centres is rotating with constant angular velocity ${\omega }_o$ . Whole situation is shown in figure. Choose the correct option(s): (Use $R=4r$)

A uniform rod $AB$ of mass $M$ is attached to a hinge at one end $A$ , and released from rest from the horizontal position. The rod rotates about $A$ , and when it reaches the vertical position the rod strikes a sphere of mass $m$ and radius $r$ initially at rest on the smooth horizontal surface as shown in the adjacent figure. The impact is along the horizontal direction and perfectly elastic. If at the moment of impact, the lowest end of the rod is very close to the smooth horizontal surface. After the impact, the sphere moves along the horizontal and the rod, subsequently rises to a maximum of $60°$ with the vertical Choose the correct statement(s) from the following, taking into account the information given above.
The length of the rod equals $\sqrt{2}r,\left(r=\frac{6\sqrt{2}}{10}\mathrm{m}\right)$
 

Two identical blocks are resting on supports as shown. Two identical bullets travelling with the same speed hit the blocks and get embedded in them in very short time. The blocks leave the contact immediately after bullet strikes them. In case $\left(\mathrm{i}\right)$ bullet's velocity is toward centre of mass and in case $\left(\mathrm{i}\mathrm{i}\right)$ bullet's velocity is off centre. The kinetic energies after collision are $K_1$ & $K_2$ and maximum height reached by the centre of mass are $h_1$ & $h_2$ respectively. Then