Motion of Connected Bodies

When two masses are joined by a light, inextensible string over a frictionless, massless pulley, analysing each mass separately (its own free body diagram) and then combining the resulting equations gives both the common acceleration and the string tension.

Case 1 — Vertical motion (Atwood-type system). Two masses $m_1>m_2$ hang from a string over a pulley; released from rest, $m_1$ accelerates down and $m_2$ accelerates up, both with the same magnitude $a$. Writing Newton's second law for each (taking the direction each mass actually moves as positive):

• For $m_2$ (up): $T-m_2g=m_2a$.
• For $m_1$ (down): $m_1g-T=m_1a$.

Adding eliminates $T$:

Substituting back:

If $m_1=m_2$, $a=0$: the system stays balanced at rest.

Case 2 — Horizontal motion. Mass $m_2$ rests on a smooth horizontal table; mass $m_1$ hangs off the table's edge via a pulley, connected by the same string, so $m_1$ falls while $m_2$ is dragged horizontally, both with acceleration $a$.

• For $m_1$ (vertical): $m_1g-T=m_1a$.
• For $m_2$ (horizontal, its only horizontal force): $T=m_2a$.

Solving simultaneously:

$a=\frac{m_1}{m_1+m_2}g,\qquad T=\frac{m_1m_2}{m_1+m_2}g.$ …