Explanation

Two particles on a horizontal surface are joined by a light inextensible string. A force P on the leading particle gives them one acceleration a and one tension T: find a from the system, then T from the trailer. Friction μN opposes motion on both — μ = 0 is smooth — and equal μ on both givesT = m₁P / (m₁ + m₂).

The string is inextensible, so the accelerations are equal. It is light, so the tension is the same throughout.

P acts on m₂, away from m₁, which keeps the string taut once the pair moves. Both particles sit on the same surface, so they share one μ. The normal reaction on each is its weight, and friction has magnitude μmg.

If P is at most the total limiting friction μ(m₁ +m₂)g, the system stays at rest.

• If P ≤ μ m₂g, m₂ is held by its own friction and T = 0.
• If μ m₂g < P ≤ μ(m₁ + m₂)g, the string tautens and T = P − μ m₂g.

Taking the direction of P as positive and assuming the pair moves:

• For the system: P − μ(m₁ + m₂)g = (m₁ + m₂)a
• For m₁: T − μ m₁g = m₁a
• For m₂: P − T − μ m₂g = m₂a

Solve the system equation for a, then the trailer for T. With the same μ on both, the friction terms cancel andT = m₁P / (m₁ + m₂). If μ = 0 the friction terms vanish.

Worked example

Two particles of masses 2 kg and 3 kg rest on a smooth horizontal surface, joined by a light inextensible string. A horizontal force of 10 N is applied to the 3 kg particle, away from the 2 kg particle. Find the acceleration of the pair and the tension in the string.

1. The surface is smooth, so the pair moves with the 10 N force.
2. For the system: 10 = 5a, so a = 2 m/s²
3. For the trailing 2 kg particle: T = 2 × 2 = 4 N

The surface is now rough, with coefficient of friction 0.2. The masses and the force are unchanged. Find a and T.

1. Total friction F = 0.2 × 5 × 9.81 = 9.81 N
2. 10 N > 9.81 N, so the pair still moves.
3. For the system: 10 − 9.81 = 5a, so a = 0.038 m/s²
4. T = 2 × 10 / 5 = 4 N, unchanged, because both particles share the same μ

Friction reduces the acceleration but not the tension. IfP were no greater than μ(m₁ + m₂)g, the pair would remain at rest.