Rotational Kinetic Energy
Introduced| English | Meaning |
|---|---|
| rotational kinetic energy/rəʊˈteɪʃənl kɪˈnetɪk ˈenədʒi/ | Energy due to rotation, equal to one half of the moment of inertia times angular speed squared for a rigid body about a fixed axis. |
A spinning flywheel stores usable energy
- A heavy flywheel, spun up fast, can power a machine for minutes after the motor stops.
- It isn't going anywhere — yet it clearly holds energy in its spin.
- A rotating body has rotational kinetic energy 转动动能, just as a moving one has translational.
- Swap mass for rotational inertia and speed for angular speed, and the formula follows.
The rotational kinetic energy formula
- A spinning body stores $E_k = \tfrac12 I \omega^2$.
- It is the exact twin of $\tfrac12 m v^2$, with $I$ for mass and $\omega$ for speed.
- Like all energy it is a scalar, measured in joules.
- Faster spin or larger rotational inertia both store more energy.
Spin stores energy
Change how fast the angle sweeps to feel how spin rate feeds the rotational kinetic energy.
A wheel with $I = 4\ \text{kg}\cdot\text{m}^2$ spins at $\omega = 3\ \tfrac{\text{rad}}{\text{s}}$. What is its rotational kinetic energy, in joules?
Difficulty 2/5 · 140 coins + a small bonus on first completion today
$E_k = \tfrac12 I\omega^2 = \tfrac12 (4)(9) = 18\ \text{J}$.
Which is the formula for rotational kinetic energy?
Difficulty 1/5 · 100 coins + a small bonus on first completion today
It is $\tfrac12 I\omega^2$ — the rotational twin of $\tfrac12 mv^2$.
Rotational kinetic energy is measured in ____.
Difficulty 1/5 · 100 coins + a small bonus on first completion today
It is an energy, so it is measured in joules.
It depends on ω squared
- Because of the $\omega^2$, doubling the spin rate stores four times the energy.
- This is why flywheels are spun as fast as their strength allows.
- Rotational inertia matters too, in direct proportion — double $I$, double the energy.
- Mass placed far from the axis (large $I$) makes a very effective energy store.
If a flywheel doubles its angular speed, its rotational kinetic energy becomes:
Difficulty 1/5 · 100 coins + a small bonus on first completion today
$E_k \propto \omega^2$, so doubling $\omega$ gives four times the energy.
Select all ways to increase a flywheel's stored rotational kinetic energy.
Difficulty 2/5 · 140 coins + a small bonus on first completion today
Higher $\omega$ or larger $I$ (mass at the rim) raises $\tfrac12 I\omega^2$. Colour has no effect.
Flywheels as batteries
- Some buses and race cars store braking energy in a spinning flywheel, then reuse it.
- The energy goes in as $\tfrac12 I\omega^2$ and comes back out to drive the wheels.
- No chemicals, no wear — just a very fast, well-balanced disk.
- It is a neat mechanical "battery" built on this one formula.
A spinning flywheel can be used to store energy and release it later.
Difficulty 1/5 · 100 coins + a small bonus on first completion today
The energy $\tfrac12 I\omega^2$ is stored in the spin and can drive a machine afterward.
Use rotational inertia $I$ and angular speed $\omega$ in $\tfrac12 I\omega^2$, never plain mass and linear speed. A rolling object has both kinds of kinetic energy at once (lesson 6.5) — don't forget the spinning part.
A wheel of rotational inertia $I = 4\ \text{kg}\cdot\text{m}^2$ spins at $\omega = 3\ \tfrac{\text{rad}}{\text{s}}$.
- $E_k = \tfrac12 I\omega^2 = \tfrac12 (4)(3^2) = \tfrac12 (4)(9) = 18\ \text{J}$.
Double the spin to $6\ \tfrac{\text{rad}}{\text{s}}$ and $E_k = \tfrac12(4)(36) = 72\ \text{J}$ — four times as much.
Rotational kinetic energy is $E_k = \tfrac12 I\omega^2$, the spin twin of $\tfrac12 mv^2$. It grows with the square of angular speed, so doubling $\omega$ stores four times the energy. Flywheels use it as a mechanical energy store.
Write from a rotational kinetic energy of 16 J
- Rotational inertia is 2 kg m squared and angular speed is 4 rad/s.
- Find the rotational kinetic energy.
A flywheel has rotational inertia 2 kg m squared and spins at 4 rad/s. What is its rotational kinetic energy?
One half of 2 is 1, and 1 times 16 is 16.
What is the rotational kinetic energy?
Difficulty 1/5 · 100 coins + a small bonus on first completion today
Half of 2 times 4 squared is 16 J.