In the above problem, let $t_1$and $t$, be the times when pure rolling of solid sphere and of hollow sphere is started. Then

A solid sphere rests on a horizontal surface. A horizontal impulse is applied at height$h$ from centre. The sphere starts rolling just after the application of impulse. The ratio $h/r$ will be

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A thin uniform rod of mass $m$ and length $l$ is kept on a smooth horizontal surface such that it can move freely. At what distance from centre of rod should a particle of mass $m$ strike on the rod such that the point $P$ at a distance $l/3$ from the end of the rod is instantaneously at rest just after the elastic collision?

In the figure shown a ring $A$ is rolling without sliding with a velocity $v$ on the horizontal surface of the body $B$ (of the same mass as $A$). All surfaces are smooth. $B$ has no initial velocity. What will be the maximum height (from initial position) reached by $A$ on $B$.

A uniform rod of mass $m$, length $l$ is placed over a smooth horizontal surface along $y$-axis and is at rest as shown in the figure. An impulsive force $F$ is applied for a small-time $∆t$ along $x$-direction at the point $A$ after this rod moves freely. The $x$–coordinate of the end $A$ of the rod when the rod becomes parallel to $x$–axis for the first time is (initially the coordinate of centre of mass of the rod is $\left(0, 0\right)$

A mass $m$ is moving at speed $v$ perpendicular to a rod of length $d$ and mass $M=6m$ which pivots around a frictionless axle running through its centre. It strikes and sticks to the end of the rod. The moment of inertia of the rod about its centre is $\frac{M{d}^2}{12}$ . Then the angular speed of the system just after the collision is

Two cylinders having radii $2R$ and $R$ and moment of inertia $4I$ and $I$ about their central axes are supported by axles perpendicular to their planes. The large cylinder is initially rotating clockwise with angular velocity ${\omega }_0.$ The small cylinder is moved to the right until it touches the large cylinder and is caused to rotate by the frictional force between the two. Eventually slipping ceases and the two cylinders rotate at constant rates in opposite directions. During this