1.A block of mass $m_1$ rests on top of mass $m_2$ which sits on a smooth floor. Coefficient of friction between blocks is $\mu$. Find maximum horizontal force applied to $m_2$ so both blocks move together without slipping.
Sol: Max friction on $m_1$: $f_{max} = \mu m_1 g \implies a_{max} = \mu g$. For both blocks: $F_{max} = (m_1 + m_2) a_{max} = \mu (m_1 + m_2) g$.
2.A bead of mass $m$ slides on a smooth wire bent into a vertical circle of radius $R$ rotating about vertical diameter with angular velocity $\omega$. Find equilibrium angle $\theta$.
Sol: $N \cos\theta = mg$, $N \sin\theta = m \omega^2 (R\sin\theta) \implies \cos\theta = \frac{g}{\omega^2 R}$ (for $\omega^2 R > g$).
