1.Define Center of Mass of a system of particles.
Sol: Point where the total mass of the system is supposed to be concentrated: $\vec{r}_{cm} = \frac{\sum m_i \vec{r}_i}{\sum m_i}$.
2.State the law of conservation of angular momentum.
Sol: If net external torque on a system is zero ($\vec{\tau}_{ext} = 0$), its total angular momentum remains constant ($\vec{L} = I\vec{\omega} = \text{const}$).
3.What is the radius of gyration? Write its formula.
Sol: The effective distance from axis of rotation where whole mass can be assumed concentrated: $K = \sqrt{\frac{I}{M}}$.
4.State the parallel axes theorem for moment of inertia.
Sol: $I = I_{cm} + M d^2$, where $d$ is perpendicular distance between parallel axes.
5.Two particles of mass $2 \text{ kg}$ and $3 \text{ kg}$ are located at $(1,2)$ and $(4,5)$. Find position of Center of Mass.
Sol: $x_{cm} = \frac{2(1)+3(4)}{5} = \frac{14}{5} = 2.8$. $y_{cm} = \frac{2(2)+3(5)}{5} = \frac{19}{5} = 3.8$. Point = $(2.8, 3.8)$.
6.Find the moment of inertia of a uniform ring of mass $M$ and radius $R$ about its diameter.
Sol: By perpendicular axis theorem, $I_{z} = I_x + I_y \implies M R^2 = 2 I_d \implies I_d = \frac{1}{2} M R^2$.
7.A wheel rotates with angular acceleration $\alpha = 3 \text{ rad/s}^2$ from rest. Find angular speed after $4 \text{ s}$.
Sol: $\omega = \omega_0 + \alpha t = 0 + 3(4) = 12 \text{ rad/s}$.
8.A solid sphere of mass $M$ and radius $R$ rolls without slipping down an inclined plane of height $h$. Find velocity at bottom.
Sol: $v = \sqrt{\frac{2gh}{1 + K^2/R^2}}$. For solid sphere, $K^2/R^2 = 2/5 \implies v = \sqrt{\frac{2gh}{7/5}} = \sqrt{\frac{10gh}{7}}$.
