Centres of mass of rigid bodiesEdexcel A-Level Further Maths: Mind map

Exam questions on Centres of mass of rigid bodies

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.
   Find the distance of the centre of mass of the solid from its plane face.2 marks
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.
   Find the distance of the centre of mass of $S$ from the plane face where the hemisphere and cylinder are joined.2 marks
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 $F$.
   Find the volume of $F$.3 marks