Centre of mass of a triangular lamina?

\frac23 of the way along a median from the vertex (\frac13 of the height from each side).

Centre of mass of a semicircular lamina?

\frac{4r}{3\pi} from the diameter.

Centre of mass of a sector with half-angle \alpha?

\frac{2r\sin\alpha}{3\alpha} from the centre.

Composite lamina moments equation?

A\bar x=\sum aixi and A\bar y=\sum aiyi.

How is a hole treated?

As a negative area (subtracted from the moment and the total area).

Centre of mass of a uniform rod in a framework?

Its midpoint, with mass proportional to length.

Framework triangle 13, 13, 10: distance of centre of mass from the 10 cm side?

\frac{13(6)+13(6)+10(0)}{36}=\frac{13}{3} cm

Condition for equilibrium of a lamina?

Zero resultant force and zero total moment about any point.

Freely suspended lamina: what is vertical?

The line from the point of suspension to the centre of mass.

How do you find the angle an edge makes with the vertical?

Use \tan\theta=\frac{horizontal offset}{vertical offset} of G from the pivot.

Centres of mass of plane figures and equilibrium of a laminaEdexcel A-Level Further Maths: Flashcards

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Where is the centre of mass of a uniform lamina with an axis of symmetry?

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Where is the centre of mass of a uniform lamina with an axis of symmetry?: On the axis of symmetry.
Centre of mass of a triangular lamina?: $\frac23$ of the way along a median from the vertex ( $\frac13$ of the height from each side).
Centre of mass of a semicircular lamina?: $\frac{4r}{3\pi}$ from the diameter.
Centre of mass of a sector with half-angle $\alpha$ ?: $\frac{2r\sin\alpha}{3\alpha}$ from the centre.
Composite lamina moments equation?: $A\bar x=\sum a_ix_i$ and $A\bar y=\sum a_iy_i$ .
How is a hole treated?: As a negative area (subtracted from the moment and the total area).
Non-uniform composite: what replaces area?: The mass of each part.
Centre of mass of a uniform rod in a framework?: Its midpoint, with mass proportional to length.
Framework triangle 13, 13, 10: distance of centre of mass from the 10 cm side?: $\frac{13(6)+13(6)+10(0)}{36}=\frac{13}{3}$ cm
Condition for equilibrium of a lamina?: Zero resultant force and zero total moment about any point.
Freely suspended lamina: what is vertical?: The line from the point of suspension to the centre of mass.
How do you find the angle an edge makes with the vertical?: Use $\tan\theta=\frac{\text{horizontal offset}}{\text{vertical offset}}$ of $G$ from the pivot.

Exam questions on Centres of mass of plane figures and equilibrium of a lamina

A uniform square lamina $OABC$ has side 12 cm, with $O$ at the origin, $A$ at $(12,0)$ and $C$ at $(0,12)$ . A square of side 4 cm is removed from the corner at $A$ , the removed square having vertices $(8,0)$ , $(12,0)$ , $(12,4)$ and $(8,4)$ . Distances are in centimetres.
The lamina is freely suspended from $O$ and hangs in equilibrium. Find the angle between $OC$ and the vertical.2 marks
A uniform lamina consists of a rectangle $ABCD$ with $AB=10$ cm and $BC=6$ cm, and a semicircle of radius 5 cm whose diameter is $AB$ and which lies outside the rectangle.
Find the distance of the centre of mass of the lamina from $CD$ .2 marks
A uniform lamina is in the shape of a right-angled triangle $ABC$ with $AB=9$ cm, $BC=12$ cm and $A\hat{B}C=90^\circ$ .
Find the distances of the centre of mass of the lamina from $AB$ and from $BC$ .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).