How are pulleys modelled at A-level?

Smooth and light – no friction, negligible mass. The pulley itself can often be ignored.

Can you treat pulley systems as one particle?

No. Particles move in different directions, so each must be considered separately.

Equation for hanging particle (mass m₂) in pulley system?

m₂g - T = m₂a (taking downward as positive for the hanging mass).

Equation for particle on smooth table (mass m₁) in pulley system?

T = m₁a (tension is the only horizontal force).

How to find acceleration without finding T first?

Add the two F=ma equations – the T terms cancel. e.g. m₂g = (m₁ + m₂)a.

What is a peg in pulley contexts?

A fixed point (e.g. nail) from which a particle can be suspended – similar to a pulley.

Block 4 kg on table, block 6 kg hanging. Find acceleration. (g=10)

6g - T = 6a, T = 4a. Add: 60 = 10a, a = 6 m/s².

When is weight of table block relevant?

Only when friction is involved – R = mg, and F = μR.

Two masses 3 kg and 5 kg hang either side of pulley. Find tension. (g=10)

5g - T = 5a, T - 3g = 3a. Add: 2g = 8a, a = 2.5. T = 3(10+2.5) = 37.5 N.

Same tension on both sides of pulley?

Yes – light string means tension is constant throughout.

Connected Bodies (Pulleys) | Pearson Edexcel International AS Maths

Connected Bodies (Pulleys)

Summary

Pulley problems involve particles connected by a light inextensible string passing over a smooth light pulley. Particles move in different directions, so always treat each particle separately—never as one system. Use F = ma for each. Common setup: one block on a horizontal table, one hanging freely.

1. What Is a Pulley?

A pulley is a wheel that rotates as a string passes over it, allowing motion of attached particles.

Modelling assumptions (A-level):

Smooth – no friction at the pulley
Light – negligible mass
Pulleys allow the string to change direction
A peg is a fixed point (e.g. nail) from which a particle can hang – similar idea

2. Key Difference from Ropes and Lifts

In pulley questions, particles move in different directions (one up, one down).

Never treat the system as one particle
Always consider each particle separately
Both have the same acceleration (a) (inextensible string)
Both experience the same tension (T) (light string)

3. Force Diagrams

Particle 1 (on table): Tension (T) horizontal (or along string). Weight only if friction involved.

Particle 2 (hanging): Tension (T) up, weight (m_2 g) down.

4. Equations of Motion

Particle 1 (moving right/up, mass (m_1)): [T - m_1 g = m_1 a\quad \text{(if hanging)}]

Or for block on smooth table: (T = m_1 a) (horizontal)

Particle 2 (hanging, mass (m_2)): [m_2 g - T = m_2 a]

Add equations to eliminate (T): (m_2 g - m_1 g = (m_1 + m_2)a) or (m_2 g = (m_1 + m_2)a) for table case.

5. Problem-Solving Steps

6. Worked Example

7. Block on Horizontal Surface

Connected Bodies (Pulleys)

Summary

1. What Is a Pulley?

A pulley is a wheel that rotates as a string passes over it, allowing motion of attached particles.

Modelling assumptions (A-level):

Smooth – no friction at the pulley
Light – negligible mass
Pulleys allow the string to change direction
A peg is a fixed point (e.g. nail) from which a particle can hang – similar idea

2. Key Difference from Ropes and Lifts

In pulley questions, particles move in different directions (one up, one down).

Never treat the system as one particle
Always consider each particle separately
Both have the same acceleration (a) (inextensible string)
Both experience the same tension (T) (light string)

3. Force Diagrams

Particle 1 (on table): Tension (T) horizontal (or along string). Weight only if friction involved.

Particle 2 (hanging): Tension (T) up, weight (m_2 g) down.

4. Equations of Motion

Particle 1 (moving right/up, mass (m_1)): [T - m_1 g = m_1 a\quad \text{(if hanging)}]

Or for block on smooth table: (T = m_1 a) (horizontal)

Particle 2 (hanging, mass (m_2)): [m_2 g - T = m_2 a]

Add equations to eliminate (T): (m_2 g - m_1 g = (m_1 + m_2)a) or (m_2 g = (m_1 + m_2)a) for table case.

5. Problem-Solving Steps

Draw diagrams – one per particle, label forces and positive direction
Write F = ma for each particle
Solve – add equations to find (a), then substitute to find (T)

For constant acceleration, suvat may be needed (e.g. velocity after release).

6. Worked Example

Block 25 kg on smooth table, block 60 kg hanging. Light inextensible string over smooth pulley. Released from rest.

Find a and T:

Table block: (T = 25a)

Hanging block: (60g - T = 60a) → (588 - T = 60a) (g=9.8)

Add: (588 = 85a) → (a = 6.92\text{ m s}^{-2})

Then (T = 25 \times 6.92 = 173) N

7. Block on Horizontal Surface

If one particle is on a smooth horizontal surface, its weight does not affect the horizontal motion. Only consider weight if friction is involved (since (F = \mu R) and (R) depends on weight).