AcademyRigid-Body Rotation
Academy
Rotational Kinetic Energy
Level 1 - Physics topic page in Rigid-Body Rotation.
Principle
Rotational kinetic energy depends on angular speed and mass distribution about the axis.
Notation
\(K_{\mathrm{rot}}\)
rotational kinetic energy
\(\mathrm{J}\)
\(I\)
moment of inertia about the axis
\(\mathrm{kg\,m^{2}}\)
\(\omega\)
angular velocity
\(\mathrm{rad\,s^{-1}}\)
\(M\)
total mass
\(\mathrm{kg}\)
\(v_{\mathrm{cm}}\)
center-of-mass speed
\(\mathrm{m\,s^{-1}}\)
\(R\)
rolling radius
\(\mathrm{m}\)
Method
Derivation 1: Add the kinetic energy of many mass elements
A rigid body rotates with one angular speed, but each mass element has speed \(v_i=r_i\omega\).
Point-mass energy
\[K_i=\frac12m_iv_i^2\]
Use rotational speed
\[v_i=r_i\omega\]
Sum all elements
\[K_{\mathrm{rot}}=\frac12\left(\sum_i m_ir_i^2\right)\omega^2\]
Define moment of inertia
\[I=\sum_i m_ir_i^2\Rightarrow K_{\mathrm{rot}}=\frac12I\omega^2\]
Derivation 2: Include translation for rolling bodies
A rolling rigid body has center-of-mass translation and rotation about the center of mass.
Split kinetic energy
\[K=\frac12Mv_{\mathrm{cm}}^2+\frac12I_{\mathrm{cm}}\omega^2\]
No-slip link
\[\omega=\frac{v_{\mathrm{cm}}}{R}\]
Write in terms of speed
\[K=\frac12\left(M+\frac{I_{\mathrm{cm}}}{R^2}\right)v_{\mathrm{cm}}^2\]
Rules
These are the compact results from the method above.
Rotational energy
\[K_{\mathrm{rot}}=\frac12I\omega^2\]
Point masses
\[I=\sum_i m_ir_i^2\]
Rolling total
\[K=\frac12Mv_{\mathrm{cm}}^2+\frac12I_{\mathrm{cm}}\omega^2\]
No-slip energy
\[K=\frac12\left(M+\frac{I_{\mathrm{cm}}}{R^2}\right)v_{\mathrm{cm}}^2\]
Examples
Question
A flywheel has
\[I=0.80\,\mathrm{kg\,m^2}\]
and \[\omega=12\,\mathrm{rad\,s^{-1}}\]
Find \[K_{\mathrm{rot}}\]
Answer
\[K_{\mathrm{rot}}=\frac12(0.80)(12^2)=57.6\,\mathrm{J}\]
Checks
- Moment of inertia depends on the chosen axis.
- Rotational kinetic energy is nonnegative.
- A rolling object's energy is not only translational.
- For the same \(M\), \(R\), and \(\omega\), mass farther out means larger energy.