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

1. 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.

   (a)

   Find the $x$-coordinate of the centre of mass of the lamina.

   [1 mark]

   • A$5.5$ cm
   • B$6.0$ cm
   • C$4.89$ cm
   • D$8.0$ cm

   (b)

   Find the $y$-coordinate of the centre of mass of the lamina.

   [1 mark]

   • A$6.0$ cm
   • B$7.0$ cm
   • C$6.5$ cm
   • D$5.78$ cm

   (c)

   The lamina is freely suspended from $O$ and hangs in equilibrium. Find the angle between $OC$ and the vertical.

   [2 marks]

   Total for question 1: 4 marks

2. 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.

   (a)

   The centre of mass of a uniform semicircular lamina of radius $r$ is $\frac{4r}{3\pi}$ from its diameter. How far is the centre of mass of the semicircle from $AB$?

   [1 mark]

   • A$3.18$ cm
   • B$2.12$ cm
   • C$3.33$ cm
   • D$2.50$ cm

   (b)

   Find the area of the lamina.

   [1 mark]

   • A$138.5$ cm²
   • B$72.5$ cm²
   • C$75.7$ cm²
   • D$99.3$ cm²

   (c)

   Find the distance of the centre of mass of the lamina from $CD$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A uniform lamina is in the shape of a right-angled triangle $ABC$ with $AB=9$ cm, $BC=12$ cm and $A\hat{B}C=90^\circ$.

   (a)

   Find the distances of the centre of mass of the lamina from $AB$ and from $BC$.

   [3 marks]

   (b)

   The lamina is freely suspended from $C$ and hangs in equilibrium. Find the angle that $BC$ makes with the vertical.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A framework is made from uniform rods, joined to form an isosceles triangle $ABC$ with $AB=AC=13$ cm and $BC=10$ cm. The framework has total mass 0.72 kg.

   (a)

   Find the distance of the centre of mass of the framework from $BC$.

   [6 marks]

   (b)

   A particle of mass 0.36 kg is attached to the framework at $C$. The framework with the particle is freely suspended from $B$ and hangs in equilibrium. Find the angle that $BC$ makes with the vertical.

   [6 marks]

   Total for question 4: 12 marks

End of questions