M10.6MechanicsStretch

Pulleys

A pulley problem connects two masses by a string over a pulley, so they accelerate at the same rate in opposite directions. Writing F = ma for each mass gives two equations you solve together for the acceleration and tension.

30 min Video by Zeeshan Zamurred Forces and Motion

Edexcel AS Level Maths: 10.6 Pulleys (Year 1)Watch the full walkthrough before the notes below.

Open on YouTube

What you'll be able to do

Set up equations for each mass over a pulley
Use the same tension and acceleration
Solve the simultaneous equations
Apply to the classic two-hanging-masses problem

The setup

For a light, smooth pulley with a light inextensible string, the $tension is the same$ throughout the string and both masses have the $same magnitude$ of acceleration. The heavier mass accelerates down, the lighter one up.

Tip — Same string ⟶ same tension; connected ⟶ same |acceleration|. These two facts unlock every pulley problem.

Equations for each mass

Apply $F = ma$ to each mass separately, taking the direction it actually moves as positive. For a hanging mass $m$ : its weight $m g$ and the tension $T$ combine to give $ma$ .

heavier: m_{1} g - T = m_{1} a, lighter: T - m_{2} g = m_{2} a

Two equations, two unknowns (

a

and

T

1Add the equations:

5 g - 3 g = (5 + 3) a

2 g = 8 a

, so

a = \frac{2 \times 9.8}{8}

Answer

a = 2.45

m/s²

Finding the tension

Once you have the acceleration, substitute it into $either$ equation to find the tension $T$ .

Formula recap

m_{1} g - T = m_{1} a

Heavier (descending) mass.

T - m_{2} g = m_{2} a

Lighter (ascending) mass.

(m_{1} - m_{2}) g = (m_{1} + m_{2}) a

Adding the equations.

Common mistakes to avoid

Using different tensions on the two sides.

A light, smooth pulley gives equal tension throughout the string.

Taking the same positive direction for both masses.

Take each mass’s direction of motion as positive (they move oppositely).

Key takeaways

Over a smooth pulley, tension is equal and accelerations have equal magnitude.
Write F = ma for each mass: m₁g − T = m₁a and T − m₂g = m₂a.
Add to find a, then substitute back for T.

Test yourself

Ready to lock in Pulleys? Pick a mode and earn XP & Dobloons.