Rotational Inertia
Introduced| English | Meaning |
|---|---|
| rotational inertia/rəʊˈteɪʃənl ɪˈnɜːʃə/ | A measure of resistance to changes in rotational motion, determined by mass and its distribution about an axis. |
A skater spins faster by pulling her arms in
- A figure skater spins slowly with arms outstretched, then pulls them in and whirls into a blur.
- She added no push — she just moved her mass closer to the axis.
- How hard something is to spin depends on where its mass sits, not only how much.
- This "rotational stubbornness" is called rotational inertia 转动惯量.
Mass distribution is everything
- Rotational inertia (moment of inertia) $I$ measures resistance to angular acceleration.
- For point masses: $I = \sum m r^2$ — each bit of mass counts by its distance squared.
- Mass far from the axis contributes far more than mass near it.
- Units are $\text{kg}\cdot\text{m}^2$.
Two $0.5\ \text{kg}$ balls sit $2\ \text{m}$ from the axis on a light rod. What is the rotational inertia, in $\text{kg}\cdot\text{m}^2$?
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$I = \sum mr^2 = 2 \times (0.5 \times 2^2) = 4\ \text{kg}\cdot\text{m}^2$.
Rotational inertia depends on:
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$I = \sum mr^2$ depends on both the mass and its distance from the axis (squared).
For point masses, rotational inertia is the sum of $m r^{\_\_}$. Fill in the power.
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$I = \sum m r^2$ — distance appears squared.
Move the same two $0.5\ \text{kg}$ balls to $1\ \text{m}$ from the axis. What is the new rotational inertia, in $\text{kg}\cdot\text{m}^2$?
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$I = 2 \times (0.5 \times 1^2) = 1\ \text{kg}\cdot\text{m}^2$ — halving $r$ quarters $I$.
Shape matters, so it has standard formulas
- The same mass can have very different $I$ depending on its shape and axis.
- A hollow ring (all mass at the rim) has more $I$ than a solid disk of the same mass.
- Standard shapes have known formulas, e.g. a solid disk is $I = \tfrac12 M R^2$.
- The $r^2$ weighting is why hollow objects feel "heavier" to spin.
Two objects have the same mass. Select all ways to make one harder to spin than the other.
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Moving mass outward (rim, hollow ring) raises $I$. Colour has no effect.
Back to the skater
- Pulling her arms in shrinks $r$, so her $I$ drops sharply (because of the $r^2$).
- With angular momentum conserved (next topic), a smaller $I$ means a larger $\omega$ — she speeds up.
- Divers and gymnasts tuck for the same reason: small $I$, fast spin.
- Extend again and the spin slows back down.
More or less rotational inertia?
Rotational inertia depends on how far the mass sits from the axis. Sort each case.
A spinning skater speeds up when she pulls her arms in because her rotational inertia decreases.
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Smaller $r$ means smaller $I$; with angular momentum conserved, $\omega$ rises.
Rotational inertia is not just mass — it depends on how the mass is arranged. Two objects of equal mass can be very different to spin. Because of the $r^2$, mass at the rim matters far more than mass near the axis.
Two $0.5\ \text{kg}$ balls sit on a light rod, each $2\ \text{m}$ from the axis.
- $I = \sum m r^2 = 2 \times (0.5 \times 2^2) = 2 \times 2 = 4\ \text{kg}\cdot\text{m}^2$.
Move them to $1\ \text{m}$ and $I$ drops to $2 \times (0.5 \times 1) = 1\ \text{kg}\cdot\text{m}^2$ — four times smaller.
Rotational inertia $I$ is the resistance to angular acceleration: $I = \sum mr^2$. It depends on where the mass sits (distance squared), not just how much — so a skater pulling her arms in lowers $I$ and spins faster. Units: $\text{kg}\cdot\text{m}^2$.
Write from a rotational inertia of 18
- A 2 kg point mass is 3 m from the axis.
- Find its rotational inertia about that axis.
A 2 kg point mass is 3 m from an axis. What is its rotational inertia about that axis?
3 squared is 9, and 2 times 9 is 18.
What is the rotational inertia?
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Mass times distance squared is 2 times 9.