1.Find the center of mass of three particles of masses $1\text{ kg}, 2\text{ kg}, 3\text{ kg}$ placed at vertices of an equilateral triangle of side $1\text{ m}$.
Sol: Coordinates: $(0,0), (1,0), (0.5, \sqrt{3}/2)$. $X_{cm} = \frac{1(0)+2(1)+3(0.5)}{6} = \frac{3.5}{6} = \frac{7}{12}\text{ m}$. $Y_{cm} = \frac{1(0)+2(0)+3(\sqrt{3}/2)}{6} = \frac{\sqrt{3}}{4}\text{ m}$.
3.State Law of Conservation of Angular Momentum with two daily-life examples.
Sol: If $\tau_{ext} = 0$, $L = I \omega = \text{Constant}$. Examples: 1. Ballet dancer folding arms to spin faster. 2. Diver tucking body to complete flips.
4.A flywheel of moment of inertia $2\text{ kg}\cdot\text{m}^2$ rotates at $300\text{ rpm}$. Find its rotational kinetic energy.
Sol: $\omega = \frac{300 \times 2\pi}{60} = 10\pi \text{ rad/s}$. $K_{rot} = \frac{1}{2} I \omega^2 = \frac{1}{2} (2) (10\pi)^2 = 100\pi^2 \approx 986.96\text{ Joules}$.
