1.The potential energy of a two-atom system is $U(r) = \frac{a}{r^{12}} - \frac{b}{r^6}$. Find equilibrium distance $r_0$ and dissociation energy.
Sol: $\frac{dU}{dr} = -\frac{12a}{r^{13}} + \frac{6b}{r^7} = 0 \implies r_0 = \left(\frac{2a}{b}\right)^{1/6}$. Dissociation energy $E_d = -U(r_0) = \frac{b^2}{4a}$.
2.A bullet of mass $m$ penetrates a block of mass $M$ suspended by a string of length $L$ and gets embedded in it. Find minimum bullet velocity $v_0$ so block completes a vertical circle.
Sol: Combined velocity after collision $V = \frac{m v_0}{m + M}$. For completing vertical circle, $V = \sqrt{5gL} \implies v_0 = \left(\frac{m + M}{m}\right)\sqrt{5gL}$.
