Laws of Motion — Connected Bodies & Circular Dynamics (Medium)

Quick Recap — Connected Bodies & Circular Dynamics

• Atwood machine: $a=\dfrac{(m_1-m_2)g}{m_1+m_2}$, tension $T=\dfrac{2m_1m_2 g}{m_1+m_2}$.
• Block on table + hanging mass (frictionless): $a=\dfrac{m_2 g}{m_1+m_2}$, $T=\dfrac{m_1m_2 g}{m_1+m_2}$.
• Circular motion: centripetal force $\dfrac{mv^2}{r}$. Banking (no friction): $v=\sqrt{rg\tan\theta}$. Flat curve: $v_{max}=\sqrt{\mu rg}$.
• Vertical circle: minimum speed at the top $\sqrt{gr}$, at the bottom $\sqrt{5gr}$; tension at the bottom $=\dfrac{mv^2}{r}+mg$.

Worked mini-example. In an Atwood machine with 3 kg and 2 kg ($g=10$): $a=\dfrac{(3-2)10}{5}=2$ m/s$^2$ and $T=\dfrac{2(3)(2)(10)}{5}=24$ N.