Non-uniform rods and rods about to tilt

Non-uniform rods

Let the weight act at distance x from one end. Use the given information (reactions, string tensions, a balance point) with moments to find x.

On the point of tilting

If a plank is about to tilt (turn) about support D, it is just losing contact with the other support C, so R_C = 0. Take moments about D: the loads on one side balance the plank's weight on the other side.

Exam phrasing

"The plank is about to tip about D" or "on the point of turning about D" both mean the reaction at the other support is zero. You may be asked how far a person can walk, or the largest load that can be placed at an end.

Worked example

A uniform plank AB of length 6 m and mass 40 kg rests on supports C (1 m from A) and D (1.5 m from B). A man of mass 60 kg walks from D towards B. How far beyond D can he go before the plank tilts?

1. About to tilt about D, so R_C = 0.
2. Plank's weight acts 3 m from A, which is 1.5 m from D on the other side.
3. M(D): 40g × 1.5 = 60g × d.

Answer: d = 1 m beyond D (5.5 m from A)

Check yourself

1. A non-uniform rod AB of length 5 m and mass 12 kg rests on supports at A and B. The reaction at A is twice the reaction at B. Find the distance of the centre of mass from A, to 3 significant figures.
   Show answer

   1.67 m. R_A + R_B = 12g with R_A = 2R_B, so R_B = 4g. M(A): 12g × x = 4g × 5, so x = 1.67 m.

2. A uniform plank AB of length 4 m and mass 20 kg rests on supports at C, 1 m from A, and at B. Find the largest mass that can be placed at A without the plank tilting.
   Show answer

   20 kg. About to tilt about C, R_B = 0. The plank's weight acts 2 m from A, which is 1 m from C. M(C): m g × 1 = 20g × 1, so m = 20 kg.

3. A uniform plank of length 8 m and mass 30 kg overhangs a wall: it rests on a support at one end A and on a support D, 6 m from A. How far beyond D can a 45 kg person walk before the plank tilts?
   Show answer

   1.33 m. About to tilt about D: R_A = 0. Plank's weight is at 4 m from A, which is 2 m from D. 30g × 2 = 45g × d, so d = 1.33 m (3 s.f.).

4. A non-uniform rod AB of length 3 m and weight 60 N hangs horizontally from two vertical strings at A and B. The tension at A is 20 N. Find the distance of the centre of mass from A.
   Show answer

   2 m. T_B = 60 - 20 = 40 N. M(A): 60x = 40 × 3, so x = 2 m.

5. In the worked example (plank 40 kg, man 60 kg), find the reaction at D when the plank is about to tilt, to 3 significant figures.
   Show answer

   980 N. R_C = 0, so R_D carries the whole weight: 100g = 980 N.

6. A plank resting on supports C and D is on the point of tilting about D. What does this tell you?
   • The reaction at D is zero
   • The reactions are equal
   • The reaction at C is zero
   • The plank is no longer in equilibrium
   Show answer

   The reaction at C is zero. It is just about to lift off C, so C no longer pushes on it.

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Sources

• Pearson Edexcel International Advanced Level in Mathematics specification (issue 3), Unit M1
• Pearson Edexcel IAL Mechanics M1 (WME01) question paper, January 2024: front-cover instructions on g and accuracy

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