Maths › Mechanics › Connected particles and pulleys
Connected particles and pulleys
Two objects tied together are one system when you want the acceleration and two separate objects when you want the tension. Knowing which view to take, and when, is the whole trick.
Builds on Forces and Newton's laws.
IN THIS TOPIC
- Model connected objects sharing an acceleration via a light inextensible string.
- Choose between whole-system and single-particle equations deliberately.
- Analyse smooth-pulley systems, finding acceleration and tension.
WHAT YOU PROBABLY THINK
The tension on the heavy side of a pulley is larger than on the light side.
System or separate
A light inextensible string makes two accelerations equal and one tension act along it. Internal forces cancel inside a system, so treating both objects as one gives the acceleration cheaply; the tension is invisible there, and only isolating one object brings it back.
Two blocks of 4 kg and 2 kg joined by a string and pulled by 18 N: the system, mass 6 kg, gives a = 3 m s⁻². The rear block alone is pulled only by the string, so T = 2 × 3 = 6 N. Two viewpoints, two equations, everything found.
Over a smooth pulley
A smooth pulley redirects the string without changing its tension. With masses of 5 kg and 3 kg hanging, take each mass's own direction of travel as positive and write Newton's second law twice:
WORKED EXAMPLE
Acceleration and tension
For the 5 kg mass: 5g − T = 5a. For the 3 kg mass: T − 3g = 3a.
Adding kills T: 2g = 8a, so a = g/4 = 2.45 m s⁻².
Then T = 3(g + a) = 3 × 12.25 = 36.75 N, about 37 N.
The check: T sits between 3g ≈ 29 N and 5g = 49 N, as the middleman between a falling heavy side and a rising light side must.
After the string breaks or a mass hits the floor, the model changes: the freed object continues under gravity alone, with suvat taking over from the tension.
THE EXAM BIT
- State the modelling: light string, so one tension; inextensible, so one acceleration; smooth pulley, so tension unchanged around it.
- Write one equation per particle in that particle's own positive direction, and add to eliminate T.
- The system equation is legal only for objects moving in one straight line, never across a pulley.
- Multi-stage questions change the model mid-flight: re-read what happens when the string breaks.
CHECK YOURSELF
Masses of 7 kg and 3 kg hang over a smooth pulley. Write the two equations of motion and find the acceleration in terms of g.
Show a hint
Each mass gets its own direction of travel as positive.
Show the answer
7g − T = 7a and T − 3g = 3a.
Adding: 4g = 10a, so a = 0.4g ≈ 3.92 m s⁻².
System for the acceleration, single particle for the tension.
One equation per mass, own direction positive, add to eliminate T.
WORKBOOK
Printable practice for this topic: original exam-style questions with room to work, and a fully worked answer book. Free to use; please do not redistribute or sell.
CHECK YOUR PROGRESS
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- Model connected objects sharing an acceleration via a light inextensible string.
- Choose between whole-system and single-particle equations deliberately.
- Analyse smooth-pulley systems, finding acceleration and tension.
No animated video for this topic yet; these notes stand alone.