Centre of mass of a non-uniform rod with mass per unit length \rho(x)?

\bar x=\frac{\int x\rho\,dx}{\int\rho\,dx}

Uniform solid of revolution about the x-axis: \bar x?

\bar x=\frac{\int xy^2\,dx}{\int y^2\,dx}

Uniform lamina under a curve: \bar x and \bar y?

\bar x=\frac{\int xy\,dx}{\int y\,dx} and \bar y=\frac{\frac12\int y^2\,dx}{\int y\,dx}

Volume of a cone, hemisphere, cylinder?

\frac13\pi r^2h, \frac23\pi r^3, \pi r^2h

Why must you choose one reference plane for a composite body?

So every distance is measured from the same place and in the same direction.

Centres of mass of rigid bodiesEdexcel A-Level Further Maths: Flashcards

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Centre of mass formula for a continuous body?

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Centre of mass formula for a continuous body?: $\bar x=\frac{\int x\,dm}{\int dm}$
Centre of mass of a non-uniform rod with mass per unit length $\rho(x)$ ?: $\bar x=\frac{\int x\rho\,dx}{\int\rho\,dx}$
Uniform solid of revolution about the $x$ -axis: $\bar x$ ?: $\bar x=\frac{\int xy^2\,dx}{\int y^2\,dx}$
Uniform lamina under a curve: $\bar x$ and $\bar y$ ?: $\bar x=\frac{\int xy\,dx}{\int y\,dx}$ and $\bar y=\frac{\frac12\int y^2\,dx}{\int y\,dx}$
Centre of mass of a solid cone of height $h$ ?: $\frac14h$ from the base.
Centre of mass of a solid hemisphere of radius $r$ ?: $\frac38r$ from the plane face.
Centre of mass of a hemispherical shell?: $\frac12r$ from the plane face.
Centre of mass of a conical shell?: $\frac13h$ from the base.
Volume of a cone, hemisphere, cylinder?: $\frac13\pi r^2h$ , $\frac23\pi r^3$ , $\pi r^2h$
In a uniform composite solid, what weights the moments?: The volume of each part.
How is a removed part treated?: As a negative volume.
In a non-uniform composite, what weights the moments?: The mass of each part.
Why must you choose one reference plane for a composite body?: So every distance is measured from the same place and in the same direction.

Exam questions on Centres of mass of rigid bodies

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
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
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

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).