Rotational Inertia
Introduced| English |
|---|
| rotational inertia/rəʊˈteɪʃənl ɪˈnɜːʃə/ |
| parallel-axis theorem/ˈpærəlel ˈæksɪs ˈθɪərəm/ |
| distribution of mass/ˌdɪstrɪˈbjuːʃn ɒv mæs/ |
Why a skater spins faster with arms tucked in
- A spinning figure skater pulls their arms in and suddenly whirls much faster.
- They added no push -- they only changed where their mass is.
- Spinning has its own kind of "inertia," and it depends on shape, not just amount.
- This is the quantity that makes rotation feel heavy or light.
Rotational inertia
- Rotational inertia 转动惯量 $I$ is the resistance to a change in spinning -- the rotational twin of mass.
- A large $I$ means a torque produces only a small angular acceleration.
- Unlike ordinary mass, it depends on how the mass is arranged.
Select all true statements about rotational inertia.
Rotational inertia resists angular acceleration and depends on mass distribution -- shape matters, so the third is false.
Distance squared
- For point masses, $I = \sum m_i r_i^2$ -- each bit's distance from the axis is squared.
- Mass far from the axis counts far more than mass near it.
- Doubling a mass's distance quadruples its contribution to $I$.
Two $3\ \text{kg}$ masses sit $2\ \text{m}$ from an axis on a light rod. Total rotational inertia (in kg·m^2)?
$I = \sum m r^2 = 3(2)^2 + 3(2)^2 = 12 + 12 = 24\ \text{kg}\cdot\text{m}^2$.
Moving a mass twice as far from the axis changes its contribution to $I$ by a factor of...
$I \propto r^2$, so doubling $r$ multiplies the contribution by $2^2 = 4$.
Continuous bodies
- For a solid object, integrate: $I = \int r^2\,dm$.
- The parallel-axis theorem 平行轴定理, $I = I_{cm} + Md^2$, shifts a known $I$ to a parallel axis a distance $d$ away.
- Standard shapes (rod, disk, sphere) have tabulated formulas.
More or less rotational inertia?
Rotational inertia depends on how far the mass sits from the axis. Sort each case.
To shift a known rotational inertia to a parallel axis, use the ____-axis theorem.
$I = I_{cm} + Md^2$ is the parallel-axis theorem.
Why distribution matters
- The distribution of mass 质量分布 is everything: same mass, different shape, different $I$.
- Pulling the skater's arms in shrinks $r$, so $I$ drops and the spin speeds up.
- A flywheel puts its mass at the rim to make $I$ -- and its stored spin -- as large as possible.
A skater pulling their arms in decreases their rotational inertia.
Arms in means mass is closer to the axis (smaller $r$), so $I$ drops -- and the spin speeds up.
Two objects have equal mass. Which has the larger rotational inertia about its center?
The hoop's mass sits at large $r$, and $r^2$ weights it heavily, so its $I$ is larger.
Two $2\ \text{kg}$ masses sit on a light rod, each $0.5\ \text{m}$ from the axis.
- $I = \sum m r^2 = 2(0.5)^2 + 2(0.5)^2 = 1\ \text{kg}\cdot\text{m}^2$.
- Slide them out to $1\ \text{m}$ and $I$ jumps to $2(1)^2 + 2(1)^2 = 4\ \text{kg}\cdot\text{m}^2$ -- four times larger.
Rotational inertia is not just "how much mass." Two objects of equal mass can have very different $I$: a hoop (mass at the rim) resists spinning far more than a solid disk of the same mass, because the $r^2$ weights the outer mass heavily.
Rotational inertia $I$ is the resistance to angular acceleration -- $I = \sum m r^2$ (or $\int r^2\,dm$), so mass far from the axis counts most. The distribution of mass decides it, and the parallel-axis theorem shifts the axis. It is why a skater spins faster with arms pulled in.