Centres of mass of rigid bodiesEdexcel A-Level Further Maths: Subtopic test

1. 1
   The region $R$ is bounded by the curve $y=\sqrt{x}$, the $x$-axis and the line $x=4$. $R$ is rotated through $2\pi$ radians about the $x$-axis to form a uniform solid. Units are centimetres.

   (a)

   Find the volume of the solid.

   [1 mark]

   • A$\frac{16\pi}{3}$ cm³
   • B$8\pi$ cm³
   • C$\frac{64\pi}{3}$ cm³
   • D$8$ cm³

   (b)

   Which integral gives the moment of the solid about the plane $x=0$, for unit density?

   [1 mark]

   • A$\pi\int_0^4x^{\frac32}\,dx$
   • B$\pi\int_0^4x\,dx$
   • C$\pi\int_0^4x^3\,dx$
   • D$\pi\int_0^4x^2\,dx$

   (c)

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

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A uniform solid $S$ is formed by joining a solid hemisphere of radius 6 cm to a solid cylinder of radius 6 cm and height 12 cm, so that the plane face of the hemisphere coincides with a circular end of the cylinder. Both parts are made of the same material.

   (a)

   The centre of mass of a uniform solid hemisphere of radius $r$ is $\frac{3r}{8}$ from its plane face. Which is the distance of the centre of mass of the hemisphere in $S$ from the plane face?

   [1 mark]

   • A$2.25$ cm
   • B$3.00$ cm
   • C$2.55$ cm
   • D$4.50$ cm

   (b)

   Find the ratio of the mass of the hemisphere to the mass of the cylinder.

   [1 mark]

   • A$2:3$
   • B$1:6$
   • C$1:3$
   • D$3:1$

   (c)

   Find the distance of the centre of mass of $S$ from the plane face where the hemisphere and cylinder are joined.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A uniform solid cone has base radius 6 cm and height 12 cm. The part of the cone above a plane parallel to the base and 6 cm from it is removed, leaving a frustum $F$.

   (a)

   Find the volume of $F$.

   [3 marks]

   (b)

   Find the distance of the centre of mass of $F$ from its larger circular face.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A non-uniform rod $AB$ has length 2 m. The mass per unit length of the rod at a point distance $x$ metres from $A$ is $(3+x)$ kg m⁻¹.

   (a)

   Find the mass of the rod and the distance of its centre of mass from $A$.

   [6 marks]

   (b)

   A particle of mass 2 kg is attached to the rod at $A$. Find the distance from $A$ of the centre of mass of the rod and particle. A second particle of mass $m$ kg is then attached at $B$, so that the whole system balances horizontally when suspended from the midpoint of the rod. Find $m$.

   [6 marks]

   Total for question 4: 12 marks

End of questions