Atwood machine

Example

Two point masses $m_1 > 0$ and $m_2 > 0$ hang on the two ends of an inextensible string of negligible mass. The string runs over a pulley without mass and without friction. Gravity acts downward with constant acceleration $g > 0$.

Let $y$ be the height of $m_1$ above a fixed level, and let $T$ be the tension of the string. The string has constant length, so the height of $m_2$ is $c - y$ for a constant $c$; this is a holonomic constraint. Newton's second law for each mass gives $$m_1 \ddot y = T - m_1 g, \qquad -m_2 \ddot y = T - m_2 g.$$ Subtracting the second equation from the first gives $\ddot y$, and putting it back into the first equation gives $T$: $$\ddot y = \frac{m_2 - m_1}{m_1 + m_2}\, g, \qquad T = \frac{2 m_1 m_2}{m_1 + m_2}\, g.$$ Check: if $m_1 = m_2$, then $\ddot y = 0$ and $T = m_1 g$, so the masses stay in equilibrium or move at constant speed. If $m_2 \to 0$, then $\ddot y \to -g$ and $T \to 0$, so $m_1$ falls freely.