Centre of mass of laminas and composite figuresEdexcel International A Level Further Maths: Subtopic test

1. 1
   A uniform lamina is made from two rectangles joined together. Rectangle $R_1$ occupies $0\le x\le 6$ and $0\le y\le 2$, and rectangle $R_2$ occupies $0\le x\le 2$ and $2\le y\le 8$, where $x$ and $y$ are distances in cm from the axes.

   (a)

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

   [1 mark]

   • A$3$ cm
   • B$2$ cm
   • C$4$ cm
   • D$1$ cm

   (b)

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

   [1 mark]

   • A$1$ cm
   • B$5$ cm
   • C$6$ cm
   • D$3$ cm

   (c)

   Find the distance between the centre of mass of the whole lamina and the centre of mass of $R_1$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A uniform semicircular lamina of mass $6$ kg has radius $12$ cm and diameter $AB$. The point $O$ is the midpoint of $AB$. Use the result in the formulae booklet for the centre of mass of a semicircular lamina.

   (a)

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

   [1 mark]

   • A$\frac{16}{\pi}$ cm
   • B$\frac{24}{\pi}$ cm
   • C$4$ cm
   • D$6$ cm

   (b)

   A particle of mass $2$ kg is attached to the lamina at $O$. Find the distance of the centre of mass of the combined body from $AB$.

   [1 mark]

   • A$30.6$ cm
   • B$15.3$ cm
   • C$3.82$ cm
   • D$5.09$ cm

   (c)

   Instead of being attached at $O$, the particle of mass $2$ kg is attached to the lamina at the point on the curved edge on the axis of symmetry. Find the distance of the centre of mass of the combined body from $AB$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A uniform circular disc has centre $O$ and radius $12$ cm. Circular holes, each of radius $4$ cm, are cut from the disc. Take the $x$-axis through $O$ and the centre $C$ of the first hole, with $OC=6$ cm.

   (a)

   The first hole is cut out. Find the distance of the centre of mass of the remaining lamina from $O$, and state which side of $O$ it lies.

   [3 marks]

   (b)

   A second hole of radius $4$ cm is also cut out, with its centre $D$ on the perpendicular to $OC$ through $O$ and $OD=6$ cm. Find the distance of the centre of mass of the lamina with both holes from $O$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A uniform lamina $L$ is made from a rectangle and a triangle. The rectangle has vertices $(0,0)$, $(6,0)$, $(6,4)$ and $(0,4)$, and the triangle has vertices $(6,0)$, $(9,0)$ and $(6,4)$, where coordinates are in cm.

   (a)

   Find the coordinates of the centre of mass of $L$.

   [6 marks]

   (b)

   A circular hole of radius $1$ cm is cut out of the rectangular part, with its centre at the centre of the rectangle. Find the coordinates of the centre of mass of the lamina that remains.

   [6 marks]

   Total for question 4: 12 marks

End of questions