Systems of Connected Objects
Overview
For objects connected by strings, treat the SYSTEM first to find acceleration, then isolate an object to find tension. Atwood machine (two masses over pulley): a = (m₁ − m₂)g/(m₁ + m₂); T = 2m₁m₂g/(m₁ + m₂). Two blocks on a surface connected by a string, force F on one: a = F/(m₁ + m₂); T = m₂ × a. Elevator problems: person of mass m in elevator accelerating at a: N = m(g + a) upward acceleration, N = m(g − a) downward acceleration. N = 0 in free fall (weightlessness).
Connected objects — treating as a system
When multiple objects are connected by strings over pulleys or pushed together, you can treat the entire system as one mass to find the acceleration. Total net force = (sum of all driving forces) − (sum of all resistive forces). Total mass = sum of all masses. a = F_net,total / m_total. This system approach is powerful but ONLY gives the acceleration, not the tension in connecting strings.
Finding internal tensions
After finding the system acceleration, apply F = ma to ONE object alone to find the tension in the string connecting it to the rest. Draw the FBD for just that object, include the tension T as an unknown, and use the acceleration you found from the system approach. Example: for an Atwood machine (two masses over a pulley), system acceleration a = (m₁ − m₂)g / (m₁ + m₂).
- ⚠Using individual object mass instead of total system mass when finding acceleration
- ⚠Not drawing a separate FBD for each object when finding tensions