Angular Impulse

A homogeneous plank of mass $M$ and length $L$ is released gently on an inclined rolling mill, which consists of a large number of uniform horizontal cylindrical rollers. Axes of the roller are fixed parallel to each other in a plane inclined at an angle $\alpha$ with the horizontal with a separation $l\left(l<<L\right)$
between two adjacent axes. The mass of a roller is $m$, each roller can rotate about its axis without friction and acceleration due to gravity is $\mathrm{g}$. Long time after the plank is released; it acquires a steady speed $v$. If this steady speed is $v=\sqrt{\frac{M^{a}gl\sin \alpha }{bm}}$, then find $a+b$.

A constant torque acting on a uniform circular wheel changes its angular momentum from $A_0$ to $4 {\mathrm{A}}_0$ in $4 \mathrm{s}$. The magnitude of this torque is

In a mess fighting match between Bheema and Duryodhana, Bheema swings his mess to hit the target with a constant angular velocity ${\omega }_0$ After hitting target mess comes to rest and surprisingly bheema feels no impulsive reaction on his hand. Mess hits the target at point $\text{'}P\text{'}$ as shown in figure. If Bheema holds the mess at end $\mathrm{A},$ then $2\ell$ will be. $($Given: Moment of inertia of mess about $A$ is $25 \mathrm{kg} {\mathrm{m}}^2.$ Total mass is $20 \mathrm{kg}.$$)$

A uniform rod of mass $\text{'}m\text{'}$ and length $\text{'}l\text{'}$ lying on a smooth horizontal table is pivoted about one end. An impulse $\text{'}J\text{'}$ is applied to the other end. Angular speed with which rod starts rotating is

A mass $m=2 \mathrm{kg}$ is suspended through a thread wrapped around a disc-shaped pulley of mass $6 \mathrm{kg}$ as shown in the figure. Friction in the axle of the pulley is absent and there is no slipping of thread. When the mass $m$ falls through a vertical distance 2 metre, the velocity acquired by it, is $\left(g=10 \mathrm{m}/{\mathrm{s}}^2\right)$

A thin uniform rod of length $l$ and mass $m$ suspended vertically is free to rotate about a smooth horizontal axis which passes through its end $A$. $A$ point mass $m$ moving horizontally with velocity $v_0$ hits the lower end $B$ of the rod and sticks to it. After collision the rod just reaches the horizontal position. Which of the following statements is WRONG?

A solid sphere of radius $R$ rests on a horizontal surface. A horizontal impulse is applied on sphere at height $h$ from centre, as shown in figure below. The sphere starts rolling just after the application of impulse. What will be the ratio $\frac{h}{R}$ ?

Two uniform rods $\mathrm{AB}$ and$\mathrm{BC}$ of masses$1 \mathrm{kg}$ and $2 \mathrm{kg}$ respectively having lengths$2 \mathrm{m}$ and$1 \mathrm{m}$ respectively are joined to each other at $\mathrm{B}$. They can rotate freely about $\mathrm{B}$ without any friction. The assembly is kept on a smooth horizontal surface as shown in the figure. A horizontal impulse$P =10 \mathrm{N} \mathrm{s}$ is applied on the rod $\mathrm{AB}$ at a distance $0.5 \mathrm{m}$ from point $\mathrm{B}$ perpendicular to the rod.

Two uniform rods $AB$and$EC$ of masses$1 kg$ and $2 kg$ respectively having lengths$2 m$ and$1 m$ respectively are joined to each other at $B$. They can rotate freely about $B$ without any friction. The assembly is kept on a smooth horizontal surface as shown in the figure. A horizontal impulse$P =10 Ns$ is applied on the rod $AB$ at a distance $0.5 m$ from point $B$ perpendicular to the rod.

A smooth uniform disc of mass $m$ and radius $R$ is connected with a heavy cart $C\text{'}$ by an inextensible string. When the cart is pushed with a velocity $v_0$,

A smooth uniform disc of mass $m$ and radius $R$ is connected with a heavy cart $C\text{'}$ by an inextensible string. When the cart is pushed with a velocity $v_0$,

A uniform rod $AB$ of mass $2 \mathrm{kg}$ and Length $30 \mathrm{cm}$ at rest on a smooth horizontal surface. An impulse of force $0.2 \mathrm{N} \mathrm{s}$ is applied to end B. The time taken by the rod to turn through at right angles will be $\frac{\pi }{\mathrm{x}} \mathrm{s},$ where $x=$ ____.

A uniform solid sphere of mass $M$ and radius $R$ is placed on a smooth horizontal surface. It is given a horizontal impulse $J$ at a height $h$ above the centre of mass and sphere starts rolling. Then the value of $h$ and speed of centre of mass are

A small body of mass m tied to a non-stretchable thread moves over a smooth horizontal plane. The other end of the thread is being drawn into a hole O (fig) with a constant velocity. The dependence of the thread tension on the distance r between the body and the hole if at $r=r_0$ the angular velocity of the thread is equal to ${\omega }_0$ will be

The figure here shows a uniform rod of mass m and length $\mathcal{l}$ at rest, in a gravity free region. The rod is hinged at one of its ends. The rod is oriented along y - axis, in the position shown here.




Suppose that the rod is made of charged non - conducting material. The linear charge density on the rod varies with distance $x$ from the hinge as $\lambda ={\lambda }_0{x}^n,$ here ${\lambda }_0$ is a constant and n is real number. The rod is in rest position and now a uniform electric field $E_0\widehat{i}$ is suddenly switched on in the region for an instant. Value of n, for which there is no reaction at the hinge, is

A uniform rod AB of mass m and length $\mathrm{l}\mathrm{ }$ is at rest on a smooth horizontal surface. An impulse p is applied to the end B. The time taken by the rod to turn through a right angle is

A uniform rod $OA$ of mass $M$ and length $2a$ rests on a smooth table and is free to turn about a smooth pivot at its end $O$, in contact with it at a distance $b$ from $O$ is an inelastic particle of mass $m$, a horizontal blow of impulse $p$ is given to the rod at a distance $x$ from $O$ in a direction perpendicular to the rod. The resultant instantaneous angular velocity of the rod is

A uniform rod of mass $m$, length $\mathcal{l}$ rests on a smooth horizontal surface. Rod is given a sharp horizontal impulse $p$ perpendicular to the rod at a distance $\frac{\mathcal{l}}{4}$ from the centre. The angular velocity of the rod will be

A cylinder is rolling over frictionless horizontal surface with velocity ${\mathrm{v}}_0$ as shown in figure. Coefficient of friction between wall and cylinder is $\mathrm{\mu }=\frac{1}{4}$ . If the collision between cylinder and wall is completely inelastic, then kinetic energy of cylinder after collision -

A cylinder is rolling over frictionless horizontal surface with velocity ${\mathrm{v}}_0$ as shown in figure. Coefficient of friction between wall and cylinder is $\mathrm{\mu }=\frac{1}{4}$ . If the collision between cylinder and wall is completely inelastic, then kinetic energy of cylinder after collision -