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

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

1. 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
2. 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
3. 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