Angular displacement θ, angular velocity ω, angular acceleration α: Question bank

1.5.2 · D5 · Physics › Rotational Mechanics › Angular displacement θ, angular velocity ω, angular accelera

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Upar wali picture master diagram hai: ek spinning disc jisme radius $r$ hai, tangential velocity $v$ hai, tangential acceleration $a_{t}$ (path ke along), centripetal acceleration $a_{c}$ (centre ki taraf), aur axis vector $ω$ (page ke bahar). Isse dhyan mein rakho — zyaadatar traps bas is figure mein ek arrow change karke bane hain.

Do derivations jinpar traps depend karte hain

True ya false — justify karo

Ek hi rigid spinning disc par do points, ek rim par aur ek hub ke paas, ka angular velocity same hota hai.

True — ek rigid body ka har point har second mein same angle sweep karta hai, isliye

ω

shared hota hai; sirf linear speed

v = r ω

differ karti hai kyunki

r

differ karta hai.

Ek hi disc par do points ki linear speed same hoti hai.

False —

v = r ω

radius ke saath scale karta hai, isliye rim point (bada

r

) tez hota hai chahe

ω

dono ke liye identical ho.

Ek object ka angular acceleration

α = 0

ho sakta hai jabki woh phir bhi accelerate kar raha ho.

True —

α = 0

hone par spin rate constant hoti hai, lekin object phir bhi ghoomta rehta hai, isliye uska velocity vector direction change karta rehta hai: woh centripetal acceleration

a_{c} = ω^{2} r \neq = 0

hai.

Agar ek instant par

ω = 0

hai, toh us instant

α

bhi

0

hona chahiye.

False — swing ke top par ya decelerating fan ke rukne ke moment par,

ω = 0

hota hai lekin

α \neq = 0

; bilkul jaisa upar phenkI gayi ball ke peak par

v = 0

hota hai lekin

g

phir bhi act karta hai.

Negative

α

ka matlab hamesha object ka slow down hona hota hai.

False — hamare anticlockwise-positive convention mein, negative

α

body ko tabhi slow karta hai jab

ω

positive ho; agar

ω

bhi negative hai (clockwise spin), toh negative

α

usse speed up karta hai. Slowing =

α

aur

ω

ke opposite signs hona.

Equation

s = r θ

kaam karta hai chahe

θ

degrees mein ho ya radians mein.

False — yeh sirf radians mein hold karta hai, kyunki radian define isi tarah hua hai ki angle 1 par arc radius ke equal ho; degrees mein ek hidden factor

π /180

hota hai.

Angular velocity ek vector hai jo rim ki motion ki direction mein point karta hai.

False —

ω

rotation ke axis ki taraf point karta hai (right-hand rule, anticlockwise ke liye page ke bahar), motion ke plane ke perpendicular, path ke tangent ki taraf nahi.

Constant

α

ke liye, ek interval par average angular velocity arithmetic mean

\frac{1}{2} (ω_{0} + ω)

hota hai.

True — constant

α

ω (t)

ko time mein seedhi line banata hai, isliye us area (angle) ka average height endpoints ka midpoint hota hai.

Ek spinning body par kisi point ka radius double karne se uska centripetal acceleration double ho jaata hai.

True —

a_{c} = ω^{2} r

, fixed

ω

par

r

mein linear hai, isliye

r

double karne se

a_{c}

double hota hai (jabki

a_{c} = v^{2} / r

tumhe mislead kar sakta hai, kyunki

v

bhi double ho jaata hai).

Error dhundho

"Ek wheel

90°

ghoomta hai, toh

s = r θ

mein

θ = 90

plug karo."

Error:

θ

radians mein hona chahiye.

90° = π /2 \approx 1.571

rad, isliye

90

use karne se arc lagbhag

57

ke factor se overshoot ho jaata hai.

"Fan slow ho raha hai, toh

ω^{2} = ω_{0}^{2} + 2 α θ

mein

α = + 45

rad/s² set karoonga."

Error: anticlockwise-positive aur positive

ω

ke saath, slow hone ke liye negative

α

chahiye. Positive value force karne se speed increase hogi, jo "slows down" ke contradict karta hai.

"High spin par tangential acceleration

a_{t} = r α

, centripetal

a_{c} = ω^{2} r

par dominate karta hai."

Error:

a_{c}

ω^{2}

ke saath badhta hai jabki

a_{t}

ω

se independent hai; high

ω

par centripetal term dominate karta hai, tangential nahi.

"Kyunki

v = r ω

hai, axis par ek point (

r = 0

) ki speed undefined hai."

Error:

r = 0

deta hai

v = 0

, bilkul defined — axis par ek point move nahi karta. Yahan kuch bhi zero se divide nahi ho raha.

θ = ω_{0} t + \frac{1}{2} α t^{2}

kisi bhi motion ke liye kaam karta hai."

Error: yeh equation constant

α

assume karta hai. Agar

α

time ke saath vary karta hai toh yeh invalid hai; tumhe

α (t)

ko directly integrate karna hoga.

"Kyunki

a_{t}

aur

a_{c}

dono accelerations hain, total

a = a_{t} + a_{c}

hai."

Error: woh perpendicular hain (ek path ke along, ek centre ki taraf), isliye woh Pythagoras se add hote hain:

a = a_{t}^{2} + a_{c}^{2}

, simple sum se nahi.

"Constant speed se circle mein move kar raha ek body ka acceleration zero hai."

Error: constant speed ka matlab phir bhi changing direction hai, isliye

a_{c} = ω^{2} r \neq = 0

. Sirf

a_{t}

yahan zero hai.

"Frequency

f

aur angular velocity

ω

different units mein same cheez hain."

Error: woh

ω = 2 π f

se related hain, equal nahi. Ek full turn

2 π

radians hai, isliye

ω

mein extra

2 π

factor hota hai.

Why questions

Hum angular quantities kyun invent karte hain instead of spinning body ke liye sirf metres track karne ke?

Ek rigid body ke alag-alag points alag arc lengths travel karte hain, isliye koi ek "distance" poori body describe nahi kar sakti — lekin har point ek angle

θ

share karta hai, jo ek clean description deta hai.

Rotational formulas mein radian "natural" unit kyun hai?

Yeh is tarah define hua hai ki angle 1 par arc length radius ke equal ho, jo

s = r θ

ko bina kisi conversion constant ke exact banata hai; degrees har equation mein

π /180

inject kar dete.

Wheel ka rim bahut tez ghoomta hai jabki hub muskil se hilta hai, agar dono same

ω

share karte hain toh kyun?

Linear speed

v = r ω

hai; shared

ω

ko rim par ek bade

r

se multiply kiya jaata hai, jo identical angular motion ke bawajood bada

v

produce karta hai.

"Time-free" equation

ω^{2} = ω_{0}^{2} + 2 α θ

doosron se zyaada useful kyun ho sakti hai?

Jab tum

ω

ω_{0}

, aur

θ

jaante ho lekin time

t

nahi, toh yeh akela constant-

α

equation hai jo kabhi

t

mention nahi karta, isliye koi unknown dangling nahi rehta.

a_{c} = ω^{2} r

ko acceleration kyun kehte hain agar speed kabhi change nahi hoti?

Acceleration, velocity vector ke change ka rate hai, jisme direction bhi include hai; ghoomne se vector ki direction change hoti hai, isliye constant speed par bhi ek real acceleration required hai.

s = r θ

ko differentiate karne par

v = r ω

sirf tabhi kyon milta hai jab

r

constant ho?

Agar

r

change hota, toh product rule ek term

θ \frac{d r}{d t}

add karta;

r

ko fixed rakhne se woh term zero ho jaata hai, jo clean

v = r ω

deta hai.

Linear–angular dictionary

x \leftrightarrow θ

v \leftrightarrow ω

a \leftrightarrow α

kaam kyun karta hai?

Dono sets same defining relations follow karte hain (

ω = d θ / d t

v = d x / d t

ko mirror karta hai), isliye same calculus same-shaped equations produce karta hai sirf symbols swap karke.

Edge cases

Uniform Circular Motion ke liye

ω

aur

α

kya hain?

ω

constant (nonzero) hai aur

α = 0

; body steadily ghoomti hai bina spin-rate mein kisi change ke, isliye sirf centripetal acceleration rehta hai.

Jis instant ek decelerating fan momentarily ruk jaata hai,

ω

α

, aur

a_{c}

kya hain?

ω = 0

, isliye

a_{c} = ω^{2} r = 0

, lekin

α \neq = 0

(abhi bhi decelerate/reverse tendency hai), jo us instant purely tangential acceleration deta hai.

Rotation axis par exactly ek point (

r = 0

) ke liye

v

a_{t}

, aur

a_{c}

ka kya hota hai?

Teeno vanish ho jaate hain:

v = r ω = 0

a_{t} = r α = 0

a_{c} = ω^{2} r = 0

— axis point stationary rehta hai chahe body kitni bhi tez spin kare.

Exactly ek full revolution ke baad angular displacement, radians aur degrees mein kya hoga?

θ = 2 π

rad

= 360°

, kyunki circumference

s = 2 π r

deta hai

θ = s / r = 2 π

Ek wheel constant

ω

par spin karta hai lekin slip aur slow hone lagta hai — kya slow-down ke dauran constant-

α

equation abhi bhi valid hai?

Sirf tab agar slow-down constant

α

par ho; agar braking force (aur isliye

α

) vary kare, toh standard kinematic equations ab apply nahi hote aur tumhe

α (t)

integrate karna hoga.

Agar

ω

negative hai aur

α

bhi negative hai, toh kya body speed up ho rahi hai ya slow down?

Speed up ho rahi hai — same-sign

ω

aur

α

ka matlab spin magnitude badh rahi hai; "slowing down" ke liye opposite signs chahiye.

Angular displacement negative ho sakta hai, aur physically iska kya matlab hai?

Haan — negative

θ

bas hamare anticlockwise-positive convention ke andar opposite (clockwise) sense mein rotation hai; magnitude phir bhi ek genuine swept angle hai.

Connections

Uniform Circular Motion — upar $α = 0$ edge case.
Centripetal Acceleration and Force — $a_{c} = ω^{2} r$ traps yahan rehte hain.
Linear Kinematics Equations — woh dictionary jinhe ye traps exploit karte hain.
Torque and Angular Acceleration — kya cheez nonzero $α$ cause karti hai.