A-level · Moments

Non-uniform rods and tiltingExam-style questions

Beams whose weight does not act at the centre — finding the weight or the position of the centre of mass by taking moments — and "on the point of tilting" problems, where the reaction at the support about to be left is zero.

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14 questions · A-level level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 3 Level 4

1
Figure 1 (not accurately drawn)

A non-uniform rod $AB$ , of length $4a$ and weight $W$ , rests in a horizontal position on a horizontal workbench. The rod is perpendicular to the edge of the workbench, and the part $CB$ of the rod projects beyond the edge, where $AC = 3a$ and $CB = a$ , as shown in Figure 1. The centre of mass of the rod is at a distance $d$ from $A$ .

A light string is attached to the rod at $A$ and is pulled vertically upwards. When the tension in the string is $T$ , the rod is on the point of tilting about $C$ .

The string is then attached to the rod at $B$ instead and is again pulled vertically upwards. When the tension in the string is $3T$ , the rod is on the point of tilting about $A$ .

Find $d$ in terms of $a$ .

(6 marks)
Level 3 · 6 marksTry it
2
Figure 1 (not accurately drawn)

A scaffold board $AB$ has length $4$ m and mass $20$ kg. The board is placed on a horizontal platform, perpendicular to the edge of the platform, so that the end $B$ sticks out beyond the edge. The board passes over the edge at the point $C$ , and the length of board beyond the edge is $CB = 1.6$ m, as shown in Figure 1.

The board is modelled as a non-uniform rod whose centre of mass is $1.8$ m from $A$ .

A tin of paint of mass $15$ kg is put on the board at $B$ . To stop the board from tipping, a bag of sand of mass $M$ kg is put on the board at $A$ . The board stays at rest in a horizontal position. The tin and the bag are modelled as particles.
(a) Find the least possible value of $M$ .

(3 marks)

The tin and the bag of sand are taken off the board. A box of tiles is then put on the board at the point $F$ , between $C$ and $B$ , where $CF = 1.2$ m. The box is modelled as a particle, and the board, still horizontal, is now on the point of tilting about $C$ .
(b) Find the magnitude of the force exerted on the board by the platform.

(3 marks)
Level 3 · 6 marksTry it
3
A non-uniform rod $AB$ of length $4$ m rests horizontally on supports at $A$ and $B$ . The reaction at $A$ is $30$ N and at $B$ is $50$ N. Find the weight of the rod and the distance of its centre of mass from $A$ .
Level 3 · 3 marksTry it
4
A non-uniform rod $AB$ of length $5$ m and weight $60$ N balances horizontally on a single support placed $2$ m from $A$ . State the distance of its centre of mass from $A$ .
Level 3 · 2 marksTry it
5
A uniform beam $AB$ of length $6$ m and weight $100$ N rests on supports at $C$ and $D$ , where $AC = 1$ m and $AD = 4$ m. A weight $W$ hangs at $B$ . Find the value of $W$ for which the beam is on the point of tilting about $D$ .
Level 3 · 2 marksTry it

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