moment of inertia - cylinder - Numbas at mathcentre.ac.uk

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Name	Status	Author	Last Modified
moment of inertia - cylinder	Ready to use	Ugur Efem	12/03/2024 12:53
moment of inertia - cylinder-non-uniform	Ready to use	Ugur Efem	18/03/2024 14:24

Question statement

Determine the moment of inertia of the cylinder $C$ of uniform density $\rho = \var{d}$ , height $\var{h}$ , and radius $\var{a}$ about the axis passing through the centres of its circular faces.

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			Name	Type	Generated Value

d	integer	5
a	integer	2
h	number	1.5
V	number	6 pi

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We can, naturally, consider this problem in cylindricla coordinates.

We may assume that the cylinder $C$ sits in the space such that the axis rotation described in the question is the $z$ -axis.

Then $C$ can be described by

$0\leq r \leq \var{a} \quad \mbox{and } \quad 0\leq \theta\leq 2\pi \quad\mbox{and}\quad 0\leq z\leq h.$

In cylindrical coordinates, the mass element is $dM = \rho dV = \rho r\,d\theta\,dr\, dz$ . Also observe that each mass element is at the distance of $r$ to the axis of rotation. Then the moment of inertia is

$I_z = \int_C r^2 \, dM = \int_C r^2\rho\, dV =\iiint_C r^3\rho\, d\theta\,dr\, dz =\\[3mm] \int_0^{\var{h}}\int_0^{2\pi}\int_0^{\var{a}} r^3\rho\, dr\,d\theta \, dz = \rho\int_0^{\var{h}}\, dz \int_0^{2\pi}\,d\theta \int_0^{\var{a}} r^3\, dr .$

Note that for the last equality we used the fact that boundaries of the integrals are constants.

Thus we obtain

$I_z = \rho [z]_0^{\var{h}} [\theta]_0^{2\pi}\left[ \frac{r^4}{4}\right]_0^{\var{a}} = \frac{1}{2}\pi\rho\times\var{h}\times{a^4} = \simplify{1/2*pi*{d}*{h}*{a}^4}.$

Also note that $I_z = \frac{1}{2}M\var{a}^2$ where $M = \rho V$ , with $V=\pi\times\var{a}^2\times\var{h}$ .

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moment of inertia - cylinder

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moment of inertia - cylinder

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Ugur Efem 2 years ago

Ugur Efem 2 years ago

Ugur Efem 2 years ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Number entry

Advanced settings

Generated value: `integer`