Where must the centre of mass lie when a body is freely suspended?

Vertically below the point of suspension.

Block on a slope: sliding condition?

\tan\alpha=\mu (limiting friction).

At the point of toppling, where does the reaction act?

At the edge about which the body is about to rotate.

Why take moments about the toppling edge?

Both R and F pass through it, so they have no moment.

What is the moment of a force?

Force \times perpendicular distance from the pivot to its line of action.

Why must three non-parallel forces in equilibrium be concurrent?

Otherwise two of them have a resultant moment about their meeting point that the third cannot cancel.

Uniform rod, weight W, at \theta to the horizontal, foot on rough ground, top on a smooth wall: wall force?

S=\frac{W}{2\tan\theta} (moments about the foot).

Equilibrium, toppling and slidingEdexcel A-Level Further Maths: Flashcards

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State the two conditions for equilibrium of a rigid body.

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State the two conditions for equilibrium of a rigid body.: Resultant force is zero and resultant moment about any point is zero.
Where must the centre of mass lie when a body is freely suspended?: Vertically below the point of suspension.
Block width $w$ , height $h$ , on a slope: toppling angle?: $\tan\alpha=\frac wh$ .
Block on a slope: sliding condition?: $\tan\alpha=\mu$ (limiting friction).
Which occurs first on a slope when $\mu<\frac wh$ ?: Sliding.
What is the limiting friction equation?: $F=\mu R$ .
At the point of toppling, where does the reaction act?: At the edge about which the body is about to rotate.
Why take moments about the toppling edge?: Both $R$ and $F$ pass through it, so they have no moment.
What is the moment of a force?: Force $\times$ perpendicular distance from the pivot to its line of action.
Why must three non-parallel forces in equilibrium be concurrent?: Otherwise two of them have a resultant moment about their meeting point that the third cannot cancel.
Uniform rod, weight $W$ , at $\theta$ to the horizontal, foot on rough ground, top on a smooth wall: wall force?: $S=\frac{W}{2\tan\theta}$ (moments about the foot).
Direction of the force from a smooth wall?: Perpendicular to the wall.

Exam questions on Equilibrium, toppling and sliding

A uniform rectangular lamina $ABCD$ has $AB=12$ cm and $BC=8$ cm. It is freely suspended from the vertex $A$ and hangs in equilibrium.
Find the distance from $A$ to the centre of mass of the lamina.2 marks
A uniform solid block has a rectangular cross-section 0.4 m wide and 0.6 m high. It rests on a rough plane inclined at an angle $\alpha$ to the horizontal, with its 0.4 m wide face in contact with the plane and the width lying along a line of greatest slope.
Find the range of values of the coefficient of friction $\mu$ for which the block topples before it slides.2 marks
A uniform rod $AB$ , of mass 12 kg and length 4 m, has its end $A$ on rough horizontal ground and its end $B$ against a smooth vertical wall. The rod lies in a vertical plane perpendicular to the wall and is inclined at $60^\circ$ to the horizontal. Take $g=9.8$ m s⁻².
Find the magnitude of the force exerted on the rod by the wall.3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).