PHSX 220 Physics I (w/ Calculus) · Unit 2: Applying Newton, statics, fluids, energy · on Exam 2
Ch 10.1–10.3 Rotational kinematics
Study notes for Rotational kinematics in PHSX 220 Physics I (w/ Calculus) at Montana State University: the big ideas, key formulas, a worked example, common mistakes and an exam tip. The same topic has endless practice questions and flashcards in Mathub.
Big ideas
- Angle $\theta$ in radians, angular velocity $\omega = d\theta/dt$ (rad/s), angular acceleration $\alpha = d\omega/dt$ (rad/s²). One revolution is $2\pi$ rad; rpm converts by $\omega = \text{rpm}\times 2\pi/60$.
- Constant $\alpha$ obeys the same four equations as 1D kinematics with $x\to\theta$, $v\to\omega$, $a\to\alpha$.
- Link to a point at radius $r$: arc length $s = r\theta$, tangential speed $v = r\omega$, tangential acceleration $a_t = r\alpha$, centripetal acceleration $a_c = r\omega^2$. Total acceleration magnitude $\sqrt{a_t^2 + a_c^2}$.
- Every point on a rigid body shares the same $\omega$ and $\alpha$; points farther out move faster.
Key formulas
Constant angular acceleration
$$\omega = \omega_0 + \alpha t,\quad \theta = \theta_0 + \omega_0 t + \tfrac12\alpha t^2,\quad \omega^2 = \omega_0^2 + 2\alpha\,\Delta\theta$$
Linear ↔ angular
$$s = r\theta,\quad v = r\omega,\quad a_t = r\alpha,\quad a_c = r\omega^2 = \frac{v^2}{r}$$
Worked example
A wheel spins up from rest to 1200 rpm in 5.0 s. Find $\alpha$ and the number of revolutions.
Show the solution
$\omega = 1200(2\pi)/60 = 125.7$ rad/s; $\alpha = 125.7/5 = 25.1\ \text{rad/s}^2$. $\theta = \tfrac12(25.1)(25) = 314$ rad $= 50$ revolutions.
Common mistakes
- Leaving angles in degrees or revolutions inside v = rω.
- Mixing tangential and centripetal acceleration; they are perpendicular.
Exam tipTranslate the whole problem into radians first, solve with the "same" kinematics equations, and convert back to revolutions at the end.
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