1.6.3
Koi bhi cheez jo repeat hoti hai — jhoolta pendulum, ghoomta wheel, vibrate karta string — uska ek rhythm hota hai. Us rhythm ko describe karne ke teen tarike hain, aur teeno ek hi baat kehte hain, bas alag "units" mein:
Period T → "Ek pura cycle kitne time mein hota hai?" (seconds per cycle)Frequency f → "Ek second mein kitne cycles hote hain?" (cycles per second = Hz)Angular frequency ω → "Angle kitni tezi se sweep karta hai?" (radians per second)
Yeh poora subtopic sirf inhe teen ke beech convert karna hai. Yeh reciprocals aur scalings hain ek doosre ke.
Definition Teen rhythm quantities
Period T ek complete cycle ke liye laga time . Unit: seconds (s).Frequency f unit time mein complete cycles ki tadaad . Unit: hertz (Hz = s − 1 Angular frequency ω phase angle ke change ki rate . Unit: radians per second (rad/s).
Step 1 — f T
Yeh step kyun? Frequency cycles per second count karti hai; period hota hai seconds per cycle. Agar ek cycle T 1/ T hai frequency.
f = time cycles = T 1 ⟹ T = f 1
Step 2 — ω
Yeh step kyun? Oscillation, uniform circular motion ka shadow (projection) hoti hai. Ek point ko ek circle ke around constant speed se jaate hue imagine karo. Ek pura lap = oscillation ka ek pura cycle. Circle ke around jaane mein 2 π
Toh ek period T 2 π rate hai:
ω = time taken angle swept = T 2 π
Step 3 — Step 1 ke saath combine karo.
Yeh step kyun? T = 1/ f ω
ω = T 2 π = 2 π f
Simple harmonic motion x ( t ) = A cos ( ω t + ϕ )
Cosine ka argument, ( ω t + ϕ ) phase hai.
Ek pura cycle tab hota hai jab phase 2 π t T = 2 π / ω
ω andar appear karta hai; f T stopwatch se measure karte ho.
Worked example Example 1 — Period se sab kuchEk pendulum T = 0.5 s f ω
Step 1: f = 1/ T = 1/0.5 = 2 Hz Kyun? Do cycles ek second mein fit hote hain.
Step 2: ω = 2 π f = 2 π ( 2 ) = 4 π ≈ 12.57 rad/s Kyun? Har cycle 2 π 4 π
Worked example Example 2 — Angular frequency se wapasEk object ka ω = 10 rad/s T f
Step 1: T = 2 π / ω = 2 π /10 ≈ 0.628 s Kyun? Angle ko poora 2 π
Step 2: f = ω / ( 2 π ) = 10/ ( 2 π ) ≈ 1.59 Hz Kyun? Radian-rate ko radians-per-cycle se divide karo.
Worked example Example 3 — Equation se read karnaDiya gaya hai x ( t ) = 3 cos ( 8 π t ) ω f T
Step 1: A cos ( ω t ) ω = 8 π rad/s Kyun? ω t
Step 2: f = ω /2 π = 8 π /2 π = 4 Hz
Step 3: T = 1/ f = 0.25 s Kyun? Direct reciprocal.
Worked example Example 4 — Pehle Forecast, phir VerifyForecast: "Agar frequency double kar dun, toh ω T
Verify: ω = 2 π f f f ω double hoga. T = 1/ f f T half ho jayega. ✔
Common mistake Mistake 1:ω = f Kyun sahi lagta hai: dono "frequencies" hain, dono tab badhte hain jab motion tez hoti hai, dono mein per-second feel hoti hai.
Fix: yeh conversion factor 2 π ω radians /s count karta hai jabki f cycles /s count karta hai. Ek cycle = 2 π ω = 2 π f
Common mistake Mistake 2:T = 2 π f ω = 2 π / f Kyun sahi lagta hai: yaad rehta hai "ek 2 π T f ω Fix: units par anchor karo. T f reciprocal hona chahiye. 2 π ω T = 1/ f ω = 2 π / T = 2 π f
Common mistake Mistake 3: calculator ko radian mode mein bhool jaanaKyun sahi lagta hai: degrees "natural" lagte hain.
Fix: ω t radians mein hota hai. Degree mode mein cos ( ω t )
Recall Feynman: ek 12-saal ke bachche ko samjhaoEk bachche ko merry-go-round par imagine karo. Period hai ek puri spin mein kitna time lagta hai (jaise "ek go-around mein 5 second"). Frequency hai wo har second mein kitni spins complete karta hai (jaise "maine is second mein 2 spins ki!"). Yeh dono flip-sides hain: agar ek spin mein aadha second lagta hai, toh tum 2 per second karte ho. Angular frequency bas wahi cheez hai jo "turning angle per second" mein measure ki gayi hai — aur kyunki ek puri spin ek poora circle hai (2 π 2 π 2 π
2 π "Omega Wears a 2 π
ω 2 π f T "f Tlips T" (f = 1/ T
Period T Ek complete cycle mein laga time (unit: seconds).
Frequency f Unit time mein complete cycles ki tadaad (unit: Hz = s⁻¹).
Angular frequency ω Phase angle ke change ki rate (unit: rad/s).
f T f = 1/ T
ω T ω = 2 π / T
ω f ω = 2 π f
ω 2 π Kyunki ek full cycle = 2 π ω f
x = A cos ( ω t + ϕ ) t Angular frequency ω
Agar f ω T ω T
ω = 12 π f f = ω /2 π = 6
T = 0.2 ω ω = 2 π /0.2 = 10 π ≈ 31.4
Simple Harmonic Motion — ω x = A cos ( ω t ) Uniform Circular Motion — 2 π ω Phase and Phase Difference — phase = ω t + ϕ Wave Speed v = fλ — frequency, time-rhythm ko spatial wavelength se jodhti hai.Springs and Pendulums — ω = k / m T , f
converts cycles to radians
Period T seconds per cycle
Frequency f cycles per second
Factor 2pi angle per cycle
