Angular displacement θ, angular velocity ω, angular acceleration α

1.5.2 · Physics › Rotational Mechanics

1. Angular Displacement θ

Radians kyun, degrees nahin? Kyunki hum radian ko define isliye karte hain taaki arc length clean nikle: 1 radian woh angle hai jo radius ke barabar arc subtend karta hai ( $s = r \Rightarrow θ = 1$ ). Isse $s = r θ$ exactly sahi rehta hai, koi conversion factor nahin. Degrees mein $π /180$ ka ugly factor har jagah aata.

2. Angular Velocity ω

Period aur frequency. Ek full turn $2 π$ rad hai time $T$ (period) mein: $ω = \frac{2 π}{T} = 2 π f$ Kyun? $ω = Δ θ /Δ t = 2 π / T$ , aur frequency $f = 1/ T$ .

3. Angular Acceleration α

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

4. Rotational Kinematics ke Equations (constant α)

5. Worked Examples

Recall Feynman: 12-saal ke bachche ko explain karo

Ek merry-go-round imagine karo. Usmein baitha har koi har second same angle ghoomta hai — wahi hai angular velocity ω, jaise "ek second mein quarter turn." Lekin tera dost jo edge ke paas baitha hai woh zyada fast guzarta hai tere se jo centre ke paas ho, chahe tum dono ne same amount ghoomaa ho. Yeh extra speed door hone se aati hai: speed = (kitna door ho) × (kitna fast spin ho), yaani $v = r ω$ . Agar ride apni spin speed up kare, woh hai angular acceleration α. Poora game bas seedhi line mein tezi se chalte jaane ka "spinning version" hai.

Connections

Uniform Circular Motion — special case jahaan $α = 0$ , $ω$ constant.
Centripetal Acceleration and Force — $a_{c} = ω^{2} r$ companion.
Linear Kinematics Equations — same algebra, variables swap karo.
Torque and Angular Acceleration — $τ = I α$ mein yahi $α$ use hota hai.
Moment of Inertia — mass ka rotational analogue.
Rolling Motion — $v = r ω$ ko translation ke saath combine karta hai.

Flashcards

Angular displacement define karo aur uski SI unit batao.

Axis ke baare mein sweep kiya gaya angle; SI unit radian (rad),

θ = s / r

s = r θ

ke liye θ radians mein kyun hona chahiye?

Radian isliye define kiya jaata hai ki arc=radius angle 1 deta hai; isse koi bhi conversion constant nahi rehta,

s = r θ

exact ban jaata hai.

Linear speed aur angular velocity ke beech relation batao.

v = r ω

s = r θ

differentiate karne se milta hai.

a_{t} = r α

kaise derive hota hai?

v = r ω

ko time ke w.r.t. differentiate karo r constant rakh ke:

a_{t} = d v / d t = r d ω / d t = r α

Tangential aur centripetal acceleration mein kya fark hai?

a_{t} = r α

speed (magnitude) change karta hai;

a_{c} = ω^{2} r

sirf direction change karta hai; yeh perpendicular hain.

Constant-α ke teen kinematic equations likho.

ω = ω_{0} + α t

;

θ = ω_{0} t + \frac{1}{2} α t^{2}

;

ω^{2} = ω_{0}^{2} + 2 α θ

ω ko period T aur frequency f se relate karo.

ω = 2 π / T = 2 π f

Ek rigid body pe alag-alag radii pe do points: kya share karte hain?

Same θ, ω, α; alag s, v,

a_{t}

(r ke saath scale hote hain).

Ek fan 30 rad/s se 10 rad mein rest tak slow hota hai. α find karo.

0 = 3 0^{2} + 2 α (10) \Rightarrow α = - 45

rad/s².

ω aur α ki units kya hain?

ω rad/s mein, α rad/s² mein.