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 Forces and Motion

Edexcel AS Level Maths: 10.6 Pulleys (Year 1)Curated by Kepler, not produced by us. Watch it 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 $\textbf{tension is the same}$ throughout the string and both masses have the $\textbf{same magnitude}$ 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 $mg$ and the tension $T$ combine to give $ma$ .

\text{heavier: } m_1 g - T = m_1 a, \quad \text{lighter: } T - m_2 g = m_2 a

Two equations, two unknowns (

a

and

T

Example 1Masses 5 kg and 3 kg over a pulley (g = 9.8). Find the acceleration.

1Add the equations:

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

2g = 8a

, so

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

Answer

a = 2.45

m/s²

Finding the tension

Once you have the acceleration, substitute it into $\textbf{either}$ 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

Answer a few questions, then review anything you missed.

Next step

Keep practising

Start with the quiz above to check the key ideas. You can also choose another topic from the practice library.

Browse practice topics

Related lessons

All 6 lessons

Put the methods into practice: see how A-Level maths duels work. An account is required to play; the overview and sample are public.