1.A solid cylinder of mass $M$ and radius $R$ is attached to a horizontal spring of spring constant $k$ and rolls without slipping on a rough horizontal surface. Find the time period of small oscillations.
Sol: Total Energy $E = \frac{1}{2} M v^2 + \frac{1}{2} I \omega^2 + \frac{1}{2} k x^2 = \frac{3}{4} M v^2 + \frac{1}{2} k x^2$. Differentiating wrt $t$: $\frac{3}{2} M v a + k x v = 0 \implies a = -\frac{2k}{3M} x \implies T = 2\pi\sqrt{\frac{3M}{2k}}$.
2.A simple pendulum of length $L$ has a bob of mass $m$ carrying charge $q$. If a uniform electric field $E$ acts vertically upwards, find the new time period of small oscillations.
Sol: Effective acceleration due to gravity $g_{eff} = g - \frac{qE}{m}$. Time period $T = 2\pi \sqrt{\frac{L}{g - qE/m}}$.
