A disc rotating about its axis with angular speed $\omega_o$ is placed lightly (without any translational push) on a perfectly frictionless table. The radius of the disc is $R$. What are the linear velocities of the points A, B and C on the disc shown in Fig. 7.41? Will the disc roll in the direction indicated ?

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 bullet of mass 10 g and speed 500 m/s is fired into a door and gets embedded exactly at the centre of the door. The door is 1.0 m wide and weighs 12 kg. It is hinged at one end and rotates about a vertical axis practically without friction. Find the angular speed of the door just after the bullet embeds into it. (Hint: The moment of inertia of the door about the vertical axis at one end is $ML^2/3$.)

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

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.

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.

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?

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.

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.

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 $\omega_1 \neq \omega_2$.

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.

Explain why friction is necessary to make the disc in Fig. 7.41 roll in the direction indicated. (a) Give the direction of frictional force at B, and the sense of frictional torque, before perfect rolling begins.

(a) Prove the theorem of perpendicular axes. (Hint : Square of the distance of a point $(x, y)$ in the $x–y$ plane from an axis through the origin and perpendicular to the plane is $x^2+y^2$).

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 : (d) Show $\frac{d L'}{dt} = \sum r'_i \times \frac{dp'_i}{dt}$ Further, show that $\frac{d L'}{dt} = \tau'_{ext}$ where $\tau'_{ext}$ is the sum of all external torques acting on the system about the centre of mass. (Hint : Use the definition of centre of mass and third law of motion. Assume the internal forces between any two particles act along the line joining the particles.)

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 ?

A solid cylinder of mass 20 kg rotates about its axis with angular speed $100 rad s^{-1}$. 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?

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 $\sum p'_i = 0$.

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