Laws of Motion — Advanced Dynamics (Hard)

Quick Recap — Advanced Dynamics (Multi-step)

• Connected on inclines/pulleys: write $ma$ for each body along its motion, add to eliminate tension: e.g. hanging $m_2$ pulling $m_1$ up a rough incline gives $a=\dfrac{m_2 g-m_1 g(\sin\theta+\mu\cos\theta)}{m_1+m_2}$.
• Vertical circle: $v_{bottom}^2=v_{top}^2+4gr$; tension bottom $\dfrac{mv_b^2}{r}+mg$, top $\dfrac{mv_t^2}{r}-mg$.
• Banking with friction: $v_{max}=\sqrt{rg\dfrac{\tan\theta+\mu}{1-\mu\tan\theta}}$.
• Chain/rope of mass: tension varies along its length; a hanging fraction slides when it exceeds what friction on the table can hold.

Worked mini-example. A block just completes a vertical circle: $v_{top}=\sqrt{gr}$, so $v_{bottom}=\sqrt{gr+4gr}=\sqrt{5gr}$ — the classic minimum-speed result.