Angular displacement θ, angular velocity ω, angular acceleration α

1. Angular Displacement θ

WHY radians, not degrees? Because we define the radian so that arc length comes out clean: 1 radian is the angle that subtends an arc equal to the radius ( $s=r \Rightarrow \theta = 1$ ). This makes $s=r\theta$ exactly true, with no conversion factor. Degrees would force an ugly $\pi/180$ everywhere.

2. Angular Velocity ω

Period and frequency. One full turn is $2\pi$ rad in time $T$ (the period): $\omega = \frac{2\pi}{T} = 2\pi f$ Why? $\omega = \Delta\theta/\Delta t = 2\pi / T$ , and frequency $f = 1/T$ .

3. Angular Acceleration α

Figure — Angular displacement θ, angular velocity ω, angular acceleration α

4. The Equations of Rotational Kinematics (constant α)

Formula Derive from

$\alpha = d\omega/dt$ (constant α) Step 1 — velocity: $\displaystyle\int_{\omega_0}^{\omega} d\omega = \int_0^t \alpha\,dt \Rightarrow \omega = \omega_0 + \alpha t.$ Why? $\alpha$ constant pulls out of the integral. Step 2 — displacement: $\displaystyle\theta = \int_0^t \omega\,dt = \int_0^t(\omega_0 + \alpha t)dt = \omega_0 t + \tfrac12\alpha t^2.$ Why? Substitute the result of Step 1. Step 3 — time-free: eliminate $t$ using $t=(\omega-\omega_0)/\alpha$ : $\omega^2 = \omega_0^2 + 2\alpha\theta.$

$\boxed{\;\omega = \omega_0 + \alpha t \quad\;\; \theta = \omega_0 t + \tfrac12\alpha t^2 \quad\;\; \omega^2 = \omega_0^2 + 2\alpha\theta\;}$

5. Worked Examples

Recall Feynman: explain to a 12-year-old

Imagine a merry-go-round. Everyone on it turns through the same angle each second — that's the angular velocity ω, like "one-quarter turn per second." But your friend sitting near the edge zooms past faster than you near the centre, even though you both turned the same amount. That extra speed comes from being far out: speed = (how far out) × (how fast it spins), i.e. $v=r\omega$ . If the ride speeds up its spin, that's angular acceleration α. The whole game is just "spinning version" of walking faster and faster in a straight line.

Connections

Uniform Circular Motion — special case where $\alpha=0$ , $\omega$ constant.
Centripetal Acceleration and Force — the $a_c = \omega^2 r$ companion.
Linear Kinematics Equations — same algebra, swap variables.
Torque and Angular Acceleration — $\tau = I\alpha$ uses this $\alpha$ .
Moment of Inertia — rotational analogue of mass.
Rolling Motion — combines $v=r\omega$ with translation.

Flashcards

Define angular displacement and its SI unit.

Angle swept about an axis; SI unit radian (rad),

\theta=s/r

Why must θ be in radians for

s=r\theta

Radian is defined so arc=radius gives angle 1; this removes any conversion constant, making

s=r\theta

exact.

State the relation between linear speed and angular velocity.

v=r\omega

, obtained by differentiating

s=r\theta

How is

a_t=r\alpha

derived?

Differentiate

v=r\omega

w.r.t. time with r constant:

a_t=dv/dt=r\,d\omega/dt=r\alpha

Difference between tangential and centripetal acceleration?

a_t=r\alpha

changes speed (magnitude);

a_c=\omega^2 r

changes direction only; they are perpendicular.

Write the three constant-α kinematic equations.

\omega=\omega_0+\alpha t

;

\theta=\omega_0 t+\tfrac12\alpha t^2

;

\omega^2=\omega_0^2+2\alpha\theta

Relate ω to period T and frequency f.

\omega=2\pi/T=2\pi f

Two points at different radii on one rigid body: what do they share?

Same θ, ω, α; different s, v,

a_t

(scale with r).

A fan slows from 30 rad/s to rest over 10 rad. Find α.

0=30^2+2\alpha(10)\Rightarrow\alpha=-45

rad/s².

Units of ω and α?

ω in rad/s, α in rad/s².

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, jab koi cheez ghoomti hai (spin karti hai) ya circle par chalti hai, tab metre me distance naapna mushkil ho jaata hai, kyunki ek hi wheel ke alag-alag points alag distance cover karte hain. Isliye hum "angle" ki bhasha use karte hain. Angular displacement θ matlab kitna angle ghooma — aur ise radian me naapte hain (degree nahi!), taaki formula $s=r\theta$ bilkul saaf bane. Radian ka definition hi aisa hai ki arc = radius ho to angle = 1.

Angular velocity ω matlab angle kitni tezi se badal raha hai, yani $\omega = d\theta/dt$ , unit rad/s. Sabse mast baat: ek rigid body ke saare points ka ω same hota hai, lekin edge waale point ki linear speed zyada hoti hai kyunki $v=r\omega$ — jitna door (bada r), utni tez. Isiliye wheel ka rim tez ghoomta dikhta hai, hub dheere.

Angular acceleration α matlab ω kitni tezi se badal raha hai ( $\alpha=d\omega/dt$ , rad/s²). Isse linear me tangential acceleration milta hai: $a_t = r\alpha$ (speed badalne wala). Yaad rakho, centripetal $a_c=\omega^2 r$ alag cheez hai — wo sirf direction modta hai, speed nahi. Dono perpendicular hote hain.

Sabse important trick: rotational kinematics linear ka exact copy hai — bas $x\to\theta,\;v\to\omega,\;a\to\alpha$ badal do. Toh $v=u+at$ ban jaata hai $\omega=\omega_0+\alpha t$ . Yahi 80/20 funda hai — ek dictionary yaad karo, poora chapter aasaan!

Angular displacement θ, angular velocity ω, angular acceleration α

1. Angular Displacement θ

2. Angular Velocity ω

3. Angular Acceleration α

4. The Equations of Rotational Kinematics (constant α)

5. Worked Examples

Connections

Flashcards

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Rotational Mechanics

Connections