Q. A rope of negligible mass is wound round a hollow cylinder of mass $3kg$ and radius $40cm$. What is the angular acceleration of the cylinder if the rope is pulled with a force of $30N$? What is the linear acceleration of the rope Assume that there is no slipping.

Q. A car weighs 1800 kg. The distance between its front and back axles is 1.8 m. Its centre of gravity is 1.05 m behind the front axle. Determine the force exerted by the level ground on each front wheel and each back wheel.

Q. A non-uniform bar of weight $W$ is suspended at rest by two strings of negligible weight as shown in fig. The angles made by the strings with the vertical are ${36.9}^o$ and ${53.1}^o$ respectively. The bar is $2m$ long. Calculate the distance d of the center of gravity of the bar from its left end.

Q. Torques of equal magnitude are applied to a hollow cylinder and a solid sphere, both having the same mass and radius. The cylinder is free to rotate about its standard axis of symmetry, and the sphere is free to rotate about an axis passing through its centre. Which of the two will acquire a greater angular speed after a given time?

Q. Find the components along the x, y, z axes of the angular momentum $l$ of a particle, whose position vector is $r$ with components x, y, z and momentum is $p$ with components $p_x,p_y$ and $p_z$. Show that if the particle moves only in the x-y plane the angular momentum has only a z-component.

Q. A hoop of radius 2 m weighs 100 kg. It rolls along a horizontal floor so that its centre of mass has a speed of 20 cm/s. How much work has to be done to stop it?

Q. A solid cylinder of mass 20 kg rotates about its axis with angular speed $100rad{s}^{-1}$. The radius of the cylinder is 0.25 m. What is the kinetic energy associated with the rotation of the cylinder? What is the magnitude of angular momentum of the cylinder about its axis?

Q. From a uniform disk of radius R, a circular hole of radius R/2 is cut out. The centre of the hole is at R/2 from the centre of the original disc. Locate the centre of gravity of the resulting flat body.

Q. A metre stick is balanced on a knife edge at its centre. When two coins, each of mass 5 g are put one on top of the other at the 12.0 cm mark, the stick is found to be balanced at 45.0 cm. What is the mass of the metre stick?

Q. A solid sphere rolls down on two different inclined planes of the same heights but different angles of inclination. (a) Will it reach the bottom with the same speed in each case? (b) Will it take longer to roll down one plane than the other? (c) If so, which one and why?

Q. Show that the area of the triangle contained between the vectors $\text{a}$ and $\text{b}$ is one half of the magnitude of $\text{a}\times \text{b}$.

Q. In the HCl molecule, the separation between the nuclei of the two atoms is about $1.27\mathring{A}(1\mathring{A}={10}^{-10}m)$. Find the approximate location of the CM of the molecule, given that a chlorine atom is about 35.5 times as massive as a hydrogen atom and nearly all the mass of an atom is concentrated in its nucleus.

Q. (a) Find the moment of inertia of a sphere about a tangent to the sphere, given the moment of inertia of the sphere about any of its diameters to be $2M{R}^2/5$, where M is the mass of the sphere and R is the radius of the sphere.
(b) Given the moment of inertia of a disc of mass M and radius R about any of its diameters to be $M{R}^2/4$, find its moment of inertia about an axis normal to the disc and passing through a point on its edge.

Q. Give the location of the centre of mass of a (i) sphere, (ii) cylinder, (iii) ring, and (iv) cube, each of uniform mass density. Does the centre of mass of a body necessarily lie inside the body ?

Q. Show that $\text{a}.(\text{b}\times \text{c})$ is equal in magnitude to the volume of the parallelopiped formed on the three vectors, $\text{a, b}$ and $\text{c}$.

Q. A child sits stationary at one end of a long trolley moving uniformly with a speed V on a smooth horizontal floor. If the child gets up and runs about on the trolley in any manner, what is the speed of the CM of the (trolley + child) system ?

Q. To maintain a rotor at a uniform angular speed of $200rad{s}^{-1}$, an engine needs to transmit a torque of 180 N m. What is the power required by the engine ? (Note: uniform angular velocity in the absence of friction implies zero torque. In practice, applied torque is needed to counter frictional torque). Assume that the engine is 100% efficient.

Q. A solid cylinder rolls up an inclined plane of angle of inclination ${30}^o$. At the bottom of the inclined plane, the centre of mass of the cylinder has a speed of $5m/s$.
1.How far will the cylinder go up the plane?
2.How long will it take to return to the bottom?

Q. Two particles, each of mass $m$ and speed $v$, travel in opposite directions along parallel lines separated by a distance $d$. Show that the vector angular momentum of the two particle system is the same whatever be the point about which the angular momentum is taken.

Q. (a) A child stands at the centre of a turntable with his two arms outstretched. The turntable is set rotating with an angular speed of 40 rev/min. How much is the angular speed of the child if he folds his hands back and thereby reduces his moment of inertia to 2/5 times the initial value ? Assume that the turntable rotates without friction.
(b) Show that the childs new kinetic energy of rotation is more than the initial kinetic energy of rotation. How do you account for this increase in kinetic energy?