S5.5 Kinetic energy of a rotating body: moment of inertia
- A rotating body stores kinetic energy.
- The amount of kinetic energy stored depends on the angular velocity and the distribution of mass around the rotation axis - the moment of inertia.
- The moment of inertia is in general different for rotations about different axes.
Let’s calculate the kinetic energy of a rigid rotating body composed of n particles, mass mi, located a distance ri from the rotation axis
- This defines the moment of inertia, which is the rotational
equivalent of mass.
Key Point 5.12
Moment of inertia for a rigid body composed of n particles mass mi a distance ri from the rotation axis is
Note: the moment of inertia depends on the rotation axis
- The units of moment of inertia are kg m2.
- Using Key Point 5.8 we can rewrite the rotational
kinetic energy of a body in a form reminiscent of Key Point 3.8
Key Point 5.13
Rotational kinetic energy of a rigid body rotating about a fixed axis is
where I is the moment of inertia about that axis.
Commentary
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The kinetic energies of translation and of rotation are not different kinds of energy, they are both kinetic energy, expressed in ways that are appropriate to the motion at hand.Example: the moment of inertia of a solid cylinder, radius R, length L, mass M about two different axes.
Note: for different rotation axes, we get different I’s.

Worked example
A solid cylinder of mass 500kg and radius 1.0m is used as a flywheel to power a lorry. If the flywheel is initially ‘charged’ by bringing it up to a speed of ω=200π rad s-1, what is its kinetic energy? If the lorry uses 8.0 kW of power, for how many minutes can it operate before the flywheel comes to rest.
Solution
![\[ K = \frac{1}{2} I \omega^2 \]](mastermathpng-3.png)
For the cylindrical flywheel
Hence
Since P=W/Δt, the lorry can operate for
![\[ \Delta t=\frac{K}{8.0 \times 10 ^3}=6.17\times 10^3\; s = 1.7\;{\rm hours} \]](mastermathpng-4.png)
Learning Resources
![]() | HRW Chapter 10.6 and 10.10. Ignore section 10.7 on calculating moments of inertia by integration. |
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