Connected Bodies - Ropes & Tow Bar (Edexcel A-Level Mathematics): Revision Notes

Here's a summary of the key steps:

1. Identify all forces: Draw separate free body diagrams for each object. Include forces like tension (in ropes), the weight of each object, normal reaction, and applied forces.
2. Apply $F = ma$ for each body: For each object, write separate equations for the net force using $F = ma$. Take care to maintain consistent acceleration for all bodies since they are connected.
3. Consider tension: For ropes or tow bars, tension acts in opposite directions on the connected objects. For example, if object A pulls on object B with tension $T$, then object B pulls on object A with the same magnitude of $T$, but in the opposite direction.

Examples:

1. A Book on a Table:

• The weight of the book acts downward.
• The table exerts an equal and opposite reaction force upward.

2. A Ball Falling Towards Earth:

• When the ball is dropped, it is pulled towards the Earth by gravity.
• Simultaneously, the Earth is pulled towards the ball with the same magnitude of force.

Given:

• Two particles P and Q of masses 5 kg and 3 kg, respectively, are connected by a piece of light inextensible string.
• Particle P is pulled by a horizontal force of 40 N along a rough horizontal plane.
• Particle P experiences a 10 N frictional force, and Q experiences a 6 N frictional force. Questions:

4. Find the acceleration of the particles.
5. Find the tension in the string.

• For Particle $P$:
• Horizontal $force = 40 \text{ N}$ to the right.
• Tension $T$ in the string acting to the left.
• Frictional $force = 10 \text{ N}$ to the left.

• For Particle $Q$:
• Tension $T$ in the string acting to the right.
• Frictional $force = 6 \text{ N}$ to the left.

For Particle $P$:

For Particle $Q$:

• The string being light is important as if it were not, it would add extra weight to the particles, affecting accelerations.
• The string being inextensible (i.e., not elastic) is important as this implies both particles accelerate at the same rate.