Atwood's Machine

The situation

Two masses $m_1 > m_2$ are connected by a light inextensible string over a smooth, massless pulley. The system is released from rest. Find the acceleration and the tension in the string.

Setting up

Since the string is inextensible and the pulley is smooth (frictionless), both masses have the same magnitude of acceleration $a$. $m_1$ (heavier) accelerates downward, $m_2$ accelerates upward.

Taking downward as positive for $m_1$ and upward as positive for $m_2$:

For $m_1$ (going down):

$m_1 g - T = m_1 a \quad \text{...(1)}$

For $m_2$ (going up):

$T - m_2 g = m_2 a \quad \text{...(2)}$

Solving for acceleration

Add equations (1) and (2):

$m_1 g - m_2 g = (m_1 + m_2)a$

$\boxed{a = \frac{(m_1 - m_2)g}{m_1 + m_2}}$

Solving for tension

From equation (2): $T = m_2(g + a)$

Substitute $a$:

$T = m_2\left(g + \frac{(m_1-m_2)g}{m_1+m_2}\right) = m_2 g \cdot \frac{m_1+m_2+m_1-m_2}{m_1+m_2} = m_2 g \cdot \frac{2m_1}{m_1+m_2}$

$\boxed{T = \frac{2m_1 m_2 g}{m_1 + m_2}}$

Checking special cases

Equal masses ($m_1 = m_2 = m$):

$a = 0, \quad T = mg$

System stays at rest, tension equals the weight of either mass. Makes sense — balanced.

One mass much larger ($m_1 \gg m_2$):

$a \approx g, \quad T \approx 2m_2 g$

$m_1$ falls nearly in free fall, $m_2$ is accelerated upward at nearly $g$.

Tension is less than either weight

$T = \frac{2m_1 m_2}{m_1 + m_2} g$

Compare with $m_2 g$:

$\frac{T}{m_2 g} = \frac{2m_1}{m_1+m_2} < 2 \quad \text{and} \quad > 1 \text{ when } m_1 > m_2/...$

More directly: $T < m_1 g$ (since $m_1$ accelerates downward — net force is downward, so $T < m_1 g$) and $T > m_2 g$ (since $m_2$ accelerates upward — net force is upward, so $T > m_2 g$).

$m_2 g < T < m_1 g$

Tension lies between the two weights. This is a useful check.

Historical note

The Atwood machine was invented by George Atwood in 1784 to verify Newton's Second Law experimentally. By using two nearly equal masses, the acceleration is much less than $g$, making it easy to measure with the instruments available at the time.