Find the center of mass of a system of three particles of masses $1\text{ kg}, 2\text{ kg}, 3\text{ kg}$ placed at vertices of an equilateral triangle of side $1\text{ m}$.
2.
State Theorem of Parallel Axes and Theorem of Perpendicular Axes. Find M.I. of a thin uniform rod about an axis perpendicular to it passing through one end.
Section B: Torque & Angular Momentum
3.
A wheel of moment of inertia $I = 2\text{ kg}\cdot\text{m}^2$ is rotating at $\omega_0 = 50\text{ rad/s}$. A constant torque $\tau = 10\text{ N}\cdot\text{m}$ brings it to rest. Find time taken and number of revolutions made.
4.
State Principle of Conservation of Angular Momentum ($L = I\omega = \text{const}$). Explain its application to a ballet dancer.
Section C: Pure Rolling Motion
5.
A solid cylinder of mass $M$ and radius $R$ rolls down an inclined plane of inclination $\theta$ without slipping. Derive its acceleration $a = \frac{2}{3} g \sin\theta$.
6.
Find total kinetic energy of a rolling sphere of mass $M$ and velocity $v$ ($K_{total} = \frac{7}{10} M v^2$).
