What does 'light' mean when modelling a rope?

The rope's mass is negligible, so tension is constant throughout its length.

What does 'inextensible' mean for a string?

The string cannot stretch, so both connected particles have the same acceleration.

When is a tow bar in tension vs thrust?

Tension when the car is accelerating (pulling); thrust when decelerating (pushing).

What is a coupling in mechanics?

A general term for the connection between two objects, simplified to a string or rod for modelling.

System equation for two particles connected by rope (horizontal)?

D - (F₁ + F₂) = (m₁ + m₂)a, where D is driving force, F₁ and F₂ are resistances.

Particle 1 equation (tow bar, horizontal)?

T - F₁ = m₁a, where T is tension, F₁ is resistance on particle 1.

Particle 2 equation (tow bar, horizontal)?

D - T - F₂ = m₂a, where D is driving force, T is tension, F₂ is resistance.

When can you treat connected particles as one system?

When both particles are travelling in the same direction; tension then cancels out.

What equation to use when there is no motion in a direction?

F = 0 (or F = ma with a = 0). Do not dismiss vertical equations in horizontal problems.

A car tows a trailer. Resistances 200 N and 100 N. Total mass 800 kg. Acceleration 2 m/s². Find driving force.

D - 300 = 800 × 2, so D = 1900 N.

Library Podcasts

Courses

Referral & Rewards

Dynamics & Statics of a Particle

Connected Bodies (Ropes & Tow Bars)

Summary

Connected bodies are two or more objects linked by ropes or tow bars. Ropes are modelled as light inextensible strings (constant tension, same acceleration). Tow bars are light rods that can be in tension or thrust. Use Newton's Second Law (F = ma) for each particle or treat the system as one when particles move in the same direction.

1. What Are Connected Bodies?

Connected particles are two or more bodies linked in some way. Common examples:

Car towing a caravan or trailer
Load raised by a lift
Two bodies connected by a rope over a pulley

Problems may involve equilibrium (stationary) or motion. For motion, use Newton's Laws.

2. Modelling Ropes

A rope is modelled as a light inextensible string:

Light: Mass negligible → tension constant throughout
Inextensible: Cannot stretch → both particles have the same acceleration (a)
A string is only in tension (never thrust)
A string can go slack if disconnected

Mathematically: (T) is the same on both sides; (a_1 = a_2 = a)

3. Modelling Tow Bars

A tow bar is modelled as a light inextensible rod:

Rod can be in tension or thrust (compression)
Car accelerating → rod in tension (pulling the trailer)
Car decelerating → rod in thrust (pushing the trailer)

A coupling is the general connection between objects (e.g. train carriages); simplified to string or rod for modelling.

4. Problem-Solving Steps

Draw diagrams – label forces, positive direction, colour-code per particle
Write equations – use (F = ma) (or (F = 0) if no motion in that direction)
Solve – may need simultaneous equations

Key insight: If both particles move in the same direction, treat as one system. Tension cancels out internally.

5. Equations for Tow Bars and Ropes

6. Worked Example

Connected Bodies (Ropes & Tow Bars)

Summary

1. What Are Connected Bodies?

Connected particles are two or more bodies linked in some way. Common examples:

Car towing a caravan or trailer
Load raised by a lift
Two bodies connected by a rope over a pulley

Problems may involve equilibrium (stationary) or motion. For motion, use Newton's Laws.

2. Modelling Ropes

A rope is modelled as a light inextensible string:

Light: Mass negligible → tension constant throughout
Inextensible: Cannot stretch → both particles have the same acceleration (a)
A string is only in tension (never thrust)
A string can go slack if disconnected

Mathematically: (T) is the same on both sides; (a_1 = a_2 = a)

3. Modelling Tow Bars

A tow bar is modelled as a light inextensible rod:

Rod can be in tension or thrust (compression)
Car accelerating → rod in tension (pulling the trailer)
Car decelerating → rod in thrust (pushing the trailer)

A coupling is the general connection between objects (e.g. train carriages); simplified to string or rod for modelling.

4. Problem-Solving Steps

Draw diagrams – label forces, positive direction, colour-code per particle
Write equations – use (F = ma) (or (F = 0) if no motion in that direction)
Solve – may need simultaneous equations

Key insight: If both particles move in the same direction, treat as one system. Tension cancels out internally.

5. Equations for Tow Bars and Ropes

System as one (horizontal motion): [D - (F_1 + F_2) = (m_1 + m_2)a]

Vertical (no motion): ((R_1 + R_2) - (m_1 + m_2)g = 0)

Particle 1: (T - F_1 = m_1 a) (horizontal)

Particle 2: (D - T - F_2 = m_2 a) (horizontal)

Where: (D) = driving force, (T) = tension, (F_1, F_2) = resistive forces, (R_1, R_2) = normal reactions.

6. Worked Example

A plane (1250 kg) tows a glider (300 kg) at 1.5 m s⁻². Resistances: plane 580 N, glider 255 N. Light inextensible tow rope.

(a) Engine force: System: (D - 580 - 255 = 1550 \times 1.5) → (D = 3070) N

(b) Tension: Glider: (T - 255 = 300 \times 1.5) → (T = 705) N