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Bodies suspended in equilibrium

Edexcel IAL Mathematics · WME03 Mechanics 3

When a body hangs freely from a point, it swings until its centre of mass is vertically below the point of suspension. That single fact turns these questions into trigonometry.

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The key fact

A body suspended from a point A hangs in equilibrium with AG vertical, where G is its centre of mass. Only two forces act (the weight at G and the force at A), so they must be in the same line.

Finding the angle

1. Find G, usually by the table method.

2. Draw the body with A and G marked, and the line AG.

3. Use a right-angled triangle with the known distances to find the angle between AG (the vertical) and the line asked about.

Exam advice

Pearson's June 2025 report says many answers to the final part of the centre of mass question were blank: candidates needed a second relevant distance and found it helpful to draw a neat triangle on the diagram. Always mark which angle is asked for (with the vertical, or with the horizontal).

Worked example

A uniform solid hemisphere of radius 4 cm is suspended from a point A on the rim of its plane face. Find the angle between the plane face and the vertical.

  1. G is on the axis, OG = 3(4)/8 = 1.5 cm from the centre O of the plane face.
  2. AG is vertical. In triangle OAG, OA = 4 (along the face) and OG = 1.5 (along the axis), with a right angle at O.
  3. Angle between the face OA and AG: tan θ = 1.5/4.

Answer: θ = 20.6°

Key idea: A suspended body hangs with its centre of mass vertically below the point of suspension; use a right-angled triangle.

Check yourself

  1. In the worked example, find the angle between the plane face and the vertical, to 1 decimal place.
    Show answer

    20.6 °. tan θ = 1.5/4, θ = 20.6°.

  2. A uniform solid cone of base radius 3 cm and height 8 cm is suspended from a point on the rim of its base. Find the angle between its axis and the vertical, to 1 decimal place.
    Show answer

    56.3 °. G is 2 cm above the base on the axis. tan θ = 3/2, θ = 56.3°.

  3. The toy from the composite lesson (hemisphere and cone of radius 3 cm, centre of mass 0.661 cm from the common face inside the cone) is suspended from a point on the rim of the common face. Find the angle between the axis and the vertical, to 1 decimal place.
    Show answer

    77.6 °. tan θ = 3/0.661, θ = 77.6°.

  4. The lamina under y = x2 from x = 0 to 2 has centre of mass (1.5, 1.2). It is suspended from the origin O. Find the angle between the straight edge along the x-axis and the vertical, to 1 decimal place.
    Show answer

    38.7 °. OG is vertical, and the angle between OG and the x-axis is given by tan θ = 1.2/1.5, so θ = 38.7°.

  5. A uniform rectangular lamina 6 cm by 4 cm is suspended from a corner. Find the angle between the longer side through that corner and the vertical, to 1 decimal place.
    Show answer

    33.7 °. G is 3 cm along and 2 cm up from the corner. tan θ = 2/3, θ = 33.7°.

  6. A rigid body is freely suspended from a point A and hangs in equilibrium. What must be true?
    • Its centre of mass is at A
    • Its longest side is vertical
    • Its axis of symmetry is horizontal
    • Its centre of mass is vertically below A
    Show answer

    Its centre of mass is vertically below A. Two forces in equilibrium must act along the same line, so G is directly below A.

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Sources

Last updated 8 November 2026. Spotted a mistake? Tell us.