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.

A cylinder of mass 10 kg and radius 15 cm is rolling perfectly on a plane of inclination $30^o$. The co-efficient of static friction $\mu_s = 0.25$. (b) What is the work done against friction during rolling ?

A man stands on a rotating platform, with his arms stretched horizontally holding a 5 kg weight in each hand. The angular speed of the platform is 30 revolutions per minute. The man then brings his arms close to his body with the distance of each weight from the axis changing from 90cm to 20cm. The moment of inertia of the man together with the platform may be taken to be constant and equal to $7.6 kg m^2$. (b) Is kinetic energy conserved in the process? If not, from where does the change come about?

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.

A cylinder of mass 10 kg and radius 15 cm is rolling perfectly on a plane of inclination $30^o$. The co-efficient of static friction $\mu_s = 0.25$. (c) If the inclination $\theta$ of the plane is increased, at what value of $\theta$ does the cylinder begin to skid, and not roll perfectly ?

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.

Two discs of moments of inertia $I_1$ and $I_2$ about their respective axes (normal to the disc and passing through the centre), and rotating with angular speeds $\omega_1$ and $\omega_2$ are brought into contact face to face with their axes of rotation coincident. (b) Show that the kinetic energy of the combined system is less than the sum of the initial kinetic energies of the two discs. How do you account for this loss in energy? Take $\omega_1 \neq \omega_2$.

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?

Read each statement below carefully, and state, with reasons, if it is true or false; (a) During rolling, the force of friction acts in the same direction as the direction of motion of the CM of the body.

Two discs of moments of inertia $I_1$ and $I_2$ about their respective axes (normal to the disc and passing through the centre), and rotating with angular speeds $\omega_1$ and $\omega_2$ are brought into contact face to face with their axes of rotation coincident. (a) What is the angular speed of the two-disc system?

Read each statement below carefully, and state, with reasons, if it is true or false; (c) The instantaneous acceleration of the point of contact during rolling is zero.

Prove the result that the velocity $v$ of translation of a rolling body (like a ring, disc, cylinder or sphere) at the bottom of an inclined plane of a height $h$ is given by $v = \sqrt{\frac{2gh}{1 + k^2/R^2}}$ using dynamical consideration (i.e. by consideration of forces and torques). Note $k$ is the radius of gyration of the body about its symmetry axis, and $R$ is the radius of the body. The body starts from rest at the top of the plane.

Give the location of the centre of mass of a (iv) cube, each of uniform mass density. Does the centre of mass of a body necessarily lie inside the body ?

To maintain a rotor at a uniform angular speed of $200 rad s^{-1}$, 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.

(b) Prove the theorem of parallel axes. (Hint : If the centre of mass of a system of $n$ particles is chosen to be the origin $\sum m_i r_i = 0$).

Read each statement below carefully, and state, with reasons, if it is true or false; (d) For perfect rolling motion, work done against friction is zero.

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

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?

A car is moving with a speed of 30 m/s on a circular path of radius 500 m. If its speed is increasing at the rate of 2 m/s, the net acceleration of the car is

Read each statement below carefully, and state, with reasons, if it is true or false; (e) A wheel moving down a perfectly frictionless inclined plane will undergo slipping (not rolling) motion.