Centres of mass of uniform bodies by integrationEdexcel International A Level Further Maths: Subtopic test

1. 1
   A uniform lamina occupies the region $R$ bounded by the curve $y=x^2$, the $x$-axis and the line $x=2$.

   (a)

   Find the area of $R$.

   [1 mark]

   • A$8$
   • B$\frac83$
   • C$4$
   • D$\frac{16}{3}$

   (b)

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

   [1 mark]

   • A$4$
   • B$\frac83$
   • C$\frac32$
   • D$\frac23$

   (c)

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

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A uniform solid $S$ is formed by attaching a solid hemisphere of radius $4$ cm to one end of a solid cylinder of radius $4$ cm and height $10$ cm. The hemisphere and the cylinder are made of the same material, and the plane face of the hemisphere coincides with an end face of the cylinder.

   (a)

   Find the distance of the centre of mass of the hemisphere from its plane face.

   [1 mark]

   • A$1.5$ cm
   • B$2.0$ cm
   • C$1.7$ cm
   • D$2.5$ cm

   (b)

   Find the ratio mass of hemisphere : mass of cylinder.

   [1 mark]

   • A$\frac{8}{15}$
   • B$\frac{2}{15}$
   • C$\frac{2}{3}$
   • D$\frac{4}{15}$

   (c)

   Find the distance of the centre of mass of $S$ from the end face of the cylinder that is not in contact with the hemisphere.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A uniform solid $S$ is formed by rotating the region bounded by the curve $y=\sqrt x$, the $x$-axis and the line $x=4$ through $2\pi$ radians about the $x$-axis. The units are centimetres.

   (a)

   Use integration to show that the centre of mass of $S$ is $\frac83$ cm from the origin $O$.

   [3 marks]

   (b)

   A uniform solid cylinder of radius $2$ cm and length $3$ cm, made of the same material as $S$, is attached to $S$ so that the two solids have the same axis and the plane face of the cylinder coincides with the plane face of $S$ at $x=4$. Find the distance of the centre of mass of the combined body from $O$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The region $R$ is bounded by the curve $y=\sin x$ for $0\le x\le\pi$ and the $x$-axis. $R$ is the shape of a uniform lamina.

   (a)

   Find the exact coordinates of the centre of mass of $R$.

   [6 marks]

   (b)

   A uniform lamina $T$ is formed by removing $R$ from the rectangle $0\le x\le\pi$, $0\le y\le1$. Find the exact $y$-coordinate of the centre of mass of $T$, and give its value to $3$ significant figures.

   [6 marks]

   Total for question 4: 12 marks

End of questions