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.

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.

The oxygen molecule has a mass of $5.30 \times 10^{-26}$ kg and a moment of inertia of $1.94 \times 10^{-46} kg m^2$ about an axis through its centre perpendicular to the lines joining the two atoms. Suppose the mean speed of such a molecule in a gas is 500 m/s and that its kinetic energy of rotation is two thirds of its kinetic energy of translation. Find the average angular velocity of the molecule.

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 $\mu_s = 0.25$. (a) How much is the force of friction acting on the cylinder ?

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

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.

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

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 ?

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 ?

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.

A solid cylinder rolls up an inclined plane of angle of inclination $30^\circ$. At the bottom of the inclined plane the centre of mass of the cylinder has a speed of 5 m/s. (a) How far will the cylinder go up the plane?

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

A solid cylinder rolls up an inclined plane of angle of inclination $30^\circ$. At the bottom of the inclined plane the centre of mass of the cylinder has a speed of 5 m/s. (b) How long will it take to return to the bottom?

Separation of Motion of a system of particles into motion of the centre of mass and motion about the centre of mass : (b) Show $K = K' + \frac{MV^2}{2}$ where $K$ is the total kinetic energy of the system of particles, $K'$ is the total kinetic energy of the system when the particle velocities are taken with respect to the centre of mass and $\frac{MV^2}{2}$ is the kinetic energy of the translation of the system as a whole (i.e. of the centre of mass motion of the system). The result has been used in Sec. 7.14.

Separation of Motion of a system of particles into motion of the centre of mass and motion about the centre of mass : (a) Show $p_i = p'_i + m_i V$ where $p_i$ is the momentum of the ith particle (of mass $m_i$) and $p'_i = m_i v'_i$. Note $v'_i$ is the velocity of the ith particle relative to the centre of mass. Also, prove using the definition of the centre of mass $\sum p'_i = 0$.

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 $7.6 kg m^2$. (a) What is his new angular speed? (Neglect friction.)

(b) Given the moment of inertia of a disc of mass $M$ and radius $R$ about any of its diameters to be $MR^2/4$, find its moment of inertia about an axis normal to the disc and passing through a point on its edge.

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?

As shown in Fig.7.40, the two sides of a step ladder BA and CA are 1.6 m long and hinged at A. A rope DE, 0.5 m is tied half way up. A weight 40 kg is suspended from a point F, 1.2 m from B along the ladder BA. Assuming the floor to be frictionless and neglecting the weight of the ladder, find the tension in the rope and forces exerted by the floor on the ladder. (Take $g = 9.8 m/s^2$) (Hint: Consider the equilibrium of each side of the ladder separately.)

Separation of Motion of a system of particles into motion of the centre of mass and motion about the centre of mass : (c) Show $L = L' + R \times MV$ where $L' = \sum r'_i \times p'_i$ is the angular momentum of the system about the centre of mass with velocities taken relative to the centre of mass. Remember $r'_i = r_i - R$; rest of the notation is the standard notation used in the chapter. Note $L'$ and $M \times R V$ can be said to be angular momenta, respectively, about and of the centre of mass of the system of particles.