Formula for \bar y of a uniform lamina under y=f(x)?

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

Why is there a \frac12 in the \bar y formula?

The strip's centre of mass is at half its height, \frac y2.

Formula for \bar x of a solid of revolution about the x-axis?

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

Formula for \bar y of a solid rotated about the y-axis?

\bar y=\frac{\int yx^2\,dy}{\int x^2\,dy}

Centre of mass of a solid hemisphere?

\frac{3r}{8} from the plane face

Centre of mass of a solid cone?

\frac h4 from the base, on the axis

Centre of mass of a semicircular lamina?

\frac{4r}{3\pi} from the diameter

Centre of mass of a triangular lamina?

\frac23 along a median from the vertex

How do you treat a hole in a composite lamina?

As a negative mass: subtract its moment.

What does symmetry tell you about a uniform body?

Its centre of mass lies on any axis or plane of symmetry.

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

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Formula for $\bar x$ of a uniform lamina under $y=f(x)$?

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Formula for $\bar x$ of a uniform lamina under $y=f(x)$ ?: $\bar x=\frac{\int xy\,dx}{\int y\,dx}$
Formula for $\bar y$ of a uniform lamina under $y=f(x)$ ?: $\bar y=\frac{\frac12\int y^2\,dx}{\int y\,dx}$
Why is there a $\frac12$ in the $\bar y$ formula?: The strip's centre of mass is at half its height, $\frac y2$ .
Formula for $\bar x$ of a solid of revolution about the $x$ -axis?: $\bar x=\frac{\int xy^2\,dx}{\int y^2\,dx}$
Formula for $\bar y$ of a solid rotated about the $y$ -axis?: $\bar y=\frac{\int yx^2\,dy}{\int x^2\,dy}$
Centre of mass of a solid hemisphere?: $\frac{3r}{8}$ from the plane face
Centre of mass of a solid cone?: $\frac h4$ from the base, on the axis
Centre of mass of a semicircular lamina?: $\frac{4r}{3\pi}$ from the diameter
Centre of mass of a hemispherical shell?: $\frac r2$ from the plane face
Centre of mass of a triangular lamina?: $\frac23$ along a median from the vertex
How do you treat a hole in a composite lamina?: As a negative mass: subtract its moment.
What does symmetry tell you about a uniform body?: Its centre of mass lies on any axis or plane of symmetry.
Volume of a solid hemisphere and of a cylinder?: $\frac23\pi r^3$ and $\pi r^2h$

Exam questions on Centres of mass of uniform bodies by integration

A uniform lamina occupies the region $R$ bounded by the curve $y=x^2$ , the $x$ -axis and the line $x=2$ .
Find the $y$ -coordinate of the centre of mass of $R$ .2 marks
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.
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
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.
Use integration to show that the centre of mass of $S$ is $\frac83$ cm from the origin $O$ .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).