All subjects · Mathematics · WME01 (M1)

Particles connected over a pulley

Two particles hanging on either side of a smooth pulley move with the same size of acceleration, one up and one down. Writing F = ma for each gives two equations to solve.

Setting up

The string is light and inextensible and the pulley is smooth, so the tension T is the same on both sides and the accelerations are equal in size. The heavier particle accelerates downwards.

Equations

For masses M (heavier) and m:

Adding gives a = (M - m)g/(M + m). Substitute back for T.

Force on the pulley and later motion

The string pulls down on the pulley with T on each side, so the force on the pulley is 2T. If the heavier particle hits the floor, the string goes slack: the lighter one then moves freely under gravity, continuing upwards with the speed it had.

Worked example

Particles of mass 5 kg and 3 kg hang over a smooth pulley and are released from rest. Find the acceleration and the tension.

  1. 5g - T = 5a and T - 3g = 3a.
  2. Adding: 2g = 8a, so a = 2.45 m/s2.
  3. T = 3(g + a) = 36.8 N.

Answer: a = 2.45 m/s2, T = 36.8 N

Key idea: Same T both sides, same size of acceleration. One F = ma for each particle, then add.

Check yourself

  1. Particles of mass 6 kg and 4 kg hang over a smooth fixed pulley. Find their acceleration, to 3 significant figures.
    Show answer

    1.96 m/s2. 6g - T = 6a and T - 4g = 4a. Adding: 2g = 10a, so a = 1.96 m/s2.

  2. For the 6 kg and 4 kg particles, find the tension in the string, to 3 significant figures.
    Show answer

    47.0 N. T = 4(9.8 + 1.96) = 47.0 N.

  3. For the 5 kg and 3 kg particles in the worked example, find the magnitude of the force exerted by the string on the pulley, to 3 significant figures.
    Show answer

    73.5 N. Two vertical tensions of 36.8 N pull down: 2T = 73.5 N.

  4. In the worked example, the 5 kg particle is 1.5 m above the floor when released. Find its speed as it hits the floor, to 3 significant figures.
    Show answer

    2.71 m/s. v2 = 0 + 2(2.45)(1.5) = 7.35, so v = 2.71 m/s.

  5. After the 5 kg particle hits the floor, the string goes slack. Find how much further the 3 kg particle rises, to 3 significant figures.
    Show answer

    0.375 m. It now moves freely: 0 = 7.35 - 2(9.8)s, so s = 0.375 m.

  6. Two particles of mass 2 kg and m kg (m > 2) hang over a smooth pulley and accelerate at 4.9 m/s2. Find m.
    Show answer

    6 kg. mg - T = 4.9m and T - 2g = 2(4.9). Adding: (m - 2)9.8 = 4.9(m + 2), so 2m - 4 = m + 2 and m = 6.

  7. Why is the tension the same on both sides of the pulley?
    • The particles have equal masses
    • The string is inextensible
    • The pulley is smooth and the string is light
    • The system is at rest
    Show answer

    The pulley is smooth and the string is light. A smooth pulley exerts no friction on the string, and a light string needs no resultant force, so T is unchanged.

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Last updated 7 November 2026. Spotted a mistake? Tell us.