Rotational Inertia
Introduced| English |
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| rotational inertia/rəʊˈteɪʃənl ɪˈnɜːʃə/ |
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 $\text{kg}\cdot\text{m}^2$?
$I = \sum mr^2 = 2 \times (0.5 \times 2^2) = 4\ \text{kg}\cdot\text{m}^2$.
Rotational inertia depends on:
$I = \sum mr^2$ 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.
$I = \sum m r^2$ — 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 $\text{kg}\cdot\text{m}^2$?
$I = 2 \times (0.5 \times 1^2) = 1\ \text{kg}\cdot\text{m}^2$ — 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.
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.
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: $\text{kg}\cdot\text{m}^2$.