Rotational Kinematics
Introduced| English | Meaning |
|---|---|
| angular position/ˈæŋɡjʊlə pəˈzɪʃn/ | angular position |
| angular velocity/ˈæŋɡjʊlə vəˈlɒsɪti/ | The rate of change of angular displacement, with a direction associated with the axis of rotation. |
| angular acceleration/ˈæŋɡjʊlə əkˌseləˈreɪʃn/ | The rate of change of angular velocity with time. |
Everything that spins plays by one set of rules
- A figure skater, a wheel, a planet, a hard drive — all rotate.
- Straight-line motion had position, velocity, acceleration; spinning has angular twins of each.
- The equations look almost identical — just swap distance for angle.
- Learn the mapping once and all of rotation opens up.
Angular quantities
- Angular position 角位置 $\theta$ is the angle turned through (in radians).
- Angular velocity 角速度 $\omega = \dfrac{d\theta}{dt}$ is how fast the angle changes.
- Angular acceleration 角加速度 $\alpha = \dfrac{d\omega}{dt}$ is how fast $\omega$ changes.
One radian is the angle whose arc length equals the radius
Angular acceleration $\alpha$ is the rotational twin of which linear quantity?
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$\alpha = d\omega/dt$ mirrors $a = dv/dt$ -- angular acceleration is the twin of linear acceleration.
The same equations, rotated
- For constant $\alpha$, the rotational equations mirror the linear ones exactly.
- Replace $x \to \theta$, $v \to \omega$, $a \to \alpha$ and every kinematic formula still works.
- Constant angular acceleration behaves just like constant linear acceleration.
Select all correct linear-to-rotational pairings.
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$x\to\theta$ and $v\to\omega$ are correct. Mass maps to rotational inertia, not angle.
When alpha is not constant
- If $\alpha$ changes with time, use calculus: $\omega = \int \alpha\,dt$ and $\theta = \int \omega\,dt$.
- Differentiate to go down the chain $\theta \to \omega \to \alpha$; integrate to go up.
- It is the same derivative chain as straight-line motion.
Rotational analogs
Each linear quantity has a rotational partner. Sort each description.
If angular acceleration varies with time, you find angular velocity by taking its ____.
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$\omega = \int \alpha\,dt$ -- integrate the angular acceleration.
A wheel starts at rest with constant $\alpha = 3\ \tfrac{\text{rad}}{\text{s}^2}$. Its angular velocity after $4\ \text{s}$ (in rad/s)?
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$\omega = \omega_0 + \alpha t = 0 + 3 \times 4 = 12\ \tfrac{\text{rad}}{\text{s}}$ -- the rotational version of $v = at$.
Period and frequency
- A rotation repeats every period $T$, and its frequency is $f = 1/T$.
- These connect to the angular velocity by $\omega = 2\pi f = \dfrac{2\pi}{T}$.
- One full turn is $2\pi$ radians, so a faster spin means a larger $\omega$.
A disk spins at $f = 4$ turns per second. What is its angular velocity (in rad/s)?
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$\omega = 2\pi f = 2\pi(4) \approx 25.13\ \tfrac{\text{rad}}{\text{s}}$.
In the formula $\omega = 2\pi f$, one full revolution counts as $2\pi$ radians.
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One turn is $2\pi$ radians, so $\omega = 2\pi f$ converts turns-per-second to radians-per-second.
A wheel spins at $f = 5$ turns per second.
- Its angular velocity is $\omega = 2\pi f = 2\pi(5) \approx 31.4\ \tfrac{\text{rad}}{\text{s}}$.
- If $\alpha$ is constant, the rotational kinematic equations then predict its later angle and spin.
Angular quantities use radians, not degrees, in these formulas. A common slip is plugging in degrees, or forgetting that one full revolution is $2\pi$ (not $360$) in the equation $\omega = 2\pi f$.
Rotation has angular position $\theta$, angular velocity $\omega = d\theta/dt$, and angular acceleration $\alpha = d\omega/dt$ — exact twins of the linear quantities. For constant $\alpha$ the kinematic equations carry over; otherwise integrate. Spin rate links to period by $\omega = 2\pi f$.
Write from an angle of 15 rad
- A wheel turns at a steady 3 rad/s for 5 s.
- Find the angle turned.
A wheel rotates at a constant 3 rad/s for 5 s. Through what angle, in radians, does it turn?
3 times 5 is 15.
Through what angle does the wheel turn?
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Angular speed times time is 3 times 5 radians.