All subjects · Mathematics · WME01 (M1)

Pulleys and pegs at the edge of a table

A common M1 set-up has one particle on a horizontal table and another hanging over a pulley or peg at the edge. The method is the same: F = ma for each particle along its own direction of motion.

Directions

The particle on the table moves horizontally and the hanging one vertically, but the string makes their accelerations equal in size. For each, write F = ma along its own direction of motion.

Smooth table

Hanging mass m, table mass M:

so a = mg/(M + m). (Friction is added in a later lesson.)

Force on the pulley

The two parts of the string are perpendicular, each with tension T. The resultant force on the pulley is sqrt(T2 + T2) = T sqrt(2), at 45° below the horizontal. A smooth peg acts exactly like a smooth pulley.

Worked example

A 4 kg block on a smooth table is joined by a string over a smooth pulley to a 1 kg hanging particle. Find a and T.

  1. Hanging: g - T = a. Block: T = 4a.
  2. Adding: g = 5a, so a = 1.96 m/s2.
  3. T = 4a = 7.84 N.

Answer: a = 1.96 m/s2, T = 7.84 N

Key idea: Each particle: F = ma along its own line of motion, same T and same size of a. Force on a corner pulley is T sqrt(2).

Check yourself

  1. A 3 kg block on a smooth table is connected over a smooth pulley at the edge to a 2 kg hanging particle. Find the acceleration, to 3 significant figures.
    Show answer

    3.92 m/s2. 2g - T = 2a and T = 3a, so 2g = 5a and a = 3.92 m/s2.

  2. For the 3 kg block and 2 kg particle, find the tension, to 3 significant figures.
    Show answer

    11.8 N. T = 3a = 11.8 N.

  3. For the 4 kg and 1 kg system in the worked example, find the magnitude of the force on the pulley, to 3 significant figures.
    Show answer

    11.1 N. T sqrt(2) = 7.84 × 1.414 = 11.1 N.

  4. In the worked example the system starts from rest. Find the time for the hanging particle to fall 0.8 m, to 3 significant figures.
    Show answer

    0.904 s. 0.8 = (1/2)(1.96)t2, so t = 0.904 s.

  5. At what angle below the horizontal does the resultant force on a corner pulley act, when one string part is horizontal and the other vertical?
    Show answer

    45 °. Equal perpendicular tensions give a resultant along the diagonal, at 45°.

  6. A string passes over a fixed smooth peg instead of a pulley. How does this change the analysis?
    • The tension doubles
    • It does not: the tension is unchanged across the peg
    • Friction must be included at the peg
    • The accelerations become different
    Show answer

    It does not: the tension is unchanged across the peg. Smooth means no friction at the peg, so it behaves exactly like a smooth pulley.

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