Visual walkthrough — SI units — seven base units and all derived units

1.1.2 · D2 · Physics › Measurement, Vectors & Kinematics › SI units — seven base units and all derived units

Yeh page the parent topic ka flagship visual walkthrough hai. Yeh Dimensional Analysis ke ideas pe lean karta hai, aur physics equations Newton's Laws of Motion, Kinematics — velocity and acceleration aur Work, Energy and Power se aati hain.

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Step 1 — Teen starter bricks aur "$[\cdot ]$" ka matlab

KYA HAI. Kisi bhi recipe se pehle, hum apne teen mechanics bricks rakhte hain aur ek shorthand define karte hain.

Saare mechanics ke liye humein ye teen bricks chahiye:

Brick	Kya measure karta hai	Unit symbol
Length	kitni door	$\text{m}$ (metre)
Mass	kitna stuff / inertia	$\text{kg}$ (kilogram)
Time	kitna time	$\text{s}$ (second)

KYUN. Neeche sab kuch bas yahi teen coloured bars ka stacking aur stretching hai. Agar tum "metre, kilogram, second" ko teen physical bars ki tarah apne dimaag mein rakh sako, toh baad ka algebra sirf un bars ka bookkeeping hai.

PICTURE. Teen alag bars — magenta = metre, violet = kilogram, orange = second. Yeh overlap nahi karte: koi bhi bar doosron se nahi ban sakta. Woh independence exactly yahi hai kyun yeh base hain.

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Step 2 — Bricks ko snap karna: ek unit ki power kaisi dikhti hai

KYA HAI. Hamare recipes mein ${\text{s}}^{-2}$ aur ${\text{m}}^2$ jaisi cheezein aayengi. Chaliye dekhte hain inका matlab kya hota hai formula mein aane se pehle.

KYUN. Beginners ke liye sabse confusing cheez ${\text{s}}^{-2}$ mein minus sign hota hai. Yeh mysterious nahi hai: ${\text{s}}^{-2}$ bas $1/{\text{s}}^2$ hai, yaani ek division jo negative exponent ki tarah likha gaya hai taaki hum sab kuch ek line mein multiply kar sakein fractions stack karne ki jagah.

PICTURE. Left: do orange bars ek ${\text{s}}^2$ square bana rahe hain. Right: wahi square ek red "÷" slash ke saath aur label ${\text{s}}^{-2}$ — same object, "multiply" se "divide" mein flip hua.

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Step 3 — Velocity: pehla snap (m ÷ s)

KYA HAI. Humara pehla derived unit. Velocity jawaab deti hai "kitni door per kitne time mein?" — toh hum ek length brick ko ek time brick se divide karte hain.

$$v=\underset{\text{distance per time}}{\underbrace{\frac{d}{t}}}\Rightarrow [v]=\frac{[d]}{[t]}=\frac{\text{m}}{\text{s}}={\text{m⋅s}}^{-1}$$

Term by term:

• $d$ = distance travelled → iska unit hai $\text{m}$ (magenta brick).
• $t$ = time taken → iska unit hai $\text{s}$ (orange brick).
• Bar   ​ = "per" → yeh $\text{s}$ ko ${\text{s}}^{-1}$ bana deta hai.

KYUN yahi equation aur koi nahi? Kyunki velocity define hi hoti hai distance over time ke roop mein — yeh seedha Kinematics — velocity and acceleration se aata hai. Hum velocity ke liye koi unit choose nahi kar rahe; definition khud hume de deti hai. Yahi poora game hai: physics equation hi recipe hai.

PICTURE. Ek magenta metre-bar ek orange second-bar ke upar baitha hua, unke beech ek division bar, result label ${\text{m⋅s}}^{-1}$.

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Step 4 — Acceleration: time se dobara divide karna

KYA HAI. Acceleration jawaab deta hai "velocity khud kitni tezi se change ho rahi hai, per second?" Toh hum velocity ki unit lete hain aur time se ek baar aur divide karte hain.

$$a=\frac{\Delta v}{t}\Rightarrow [a]=\frac{[v]}{[t]}=\frac{{\text{m⋅s}}^{-1}}{\text{s}}={\text{m⋅s}}^{-2}$$

Term by term:

• $\Delta v$ = velocity mein change → unit ${\text{m⋅s}}^{-1}$ (Step 3 se).
• $t$ se dobara divide karna → ek aur ${\text{s}}^{-1}$ add hota hai, giving ${\text{s}}^{-2}$.

KYUN double "per second"? ${\text{m⋅s}}^{-2}$ ko loud mein padho "metres per second, per second". Pehla "per second" speed hai; doosra "per second" hai ki woh speed kitni tezi se badh rahi hai. Do time-divisions stacked — yahi hai jo exponent $-2$ count kar raha hai.

PICTURE. Step 3 ka ${\text{m⋅s}}^{-1}$ label ek doosre orange time-divider se guzar raha hai, ${\text{m⋅s}}^{-2}$ par land ho raha hai. Ek chhoti car neeche accelerate karti hai jo remind karti hai ki accelerate karna kaisa dikhta hai.

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Step 5 — Newton: force = mass × acceleration

KYA HAI. Ab mechanics ka star. Newton's second law kehta hai force = mass times acceleration — toh hum mass brick ko acceleration unit se multiply karte hain jo humne abhi banaya.

$$F=ma\Rightarrow [F]=[m][a]=\text{kg}\cdot {\text{m⋅s}}^{-2}=\underset{\text{define this as }1\text{N}}{\underbrace{{\text{kg⋅m⋅s}}^{-2}}}$$

Term by term:

• $m$ = mass → $\text{kg}$ (violet brick).
• $a$ = acceleration → ${\text{m⋅s}}^{-2}$ (Step 4 mein banaya).
• Dot "$\cdot$" = "times" → hum violet brick ko ${\text{m⋅s}}^{-2}$ block par stack karte hain.
• Result ko $1\text{N}$ (newton) naam diya jaata hai sirf convenience ke liye — yeh abhi bhi sirf teen bricks hain.

KYUN yeh derived hai, base nahi. Koi alag "force standard" kisi vault mein lock nahi hai. Force ka unit $F=ma$ se nikal ke aata hai (dekho Newton's Laws of Motion). Isliye parent note insist karta hai ki newton ek derived unit hai — mass, length, time sab kaam karte hain.

PICTURE. Violet $\text{kg}$ brick orange-aur-magenta ${\text{m⋅s}}^{-2}$ block par LEGO piece ki tarah click ho raha hai, ek bada banner "$=1\text{N}$" ke saath.

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Step 6 — Joule: force ek distance mein kaam karta hai

KYA HAI. Energy (work) force hai jo ek distance par apply ki jaati hai. Toh newton ko ek aur metre brick se multiply karo.

$$W=Fd\Rightarrow [W]=[F][d]=\underset{\text{newton}}{\underbrace{{\text{kg⋅m⋅s}}^{-2}}}\cdot \text{m}={\text{kg⋅m}}^2{\text{⋅s}}^{-2}\equiv 1\text{J}$$

Term by term:

• $F$ = force → ${\text{kg⋅m⋅s}}^{-2}$ (Step 5).
• $d$ = distance pushed → ek aur $\text{m}$, toh $\text{m}$ ban jaata hai ${\text{m}}^2$.
• Notice karo ki ${\text{s}}^{-2}$ nahi badla — humne ek length se multiply kiya, time se nahi.

KYUN sirf metre-count badhta hai. Length brick se multiply karna sirf length pile ko touch karta hai. Mass pile aur time pile untouched rehte hain — ek clean, visual reason ki $\text{m}$ kyun ${\text{m}}^2$ banta hai jabki $\text{kg}$ aur ${\text{s}}^{-2}$ wahi rehte hain. Yeh Work, Energy and Power se energy unit hai.

PICTURE. Newton block par ek doosra magenta metre-brick uske single metre par snap ho raha hai, ek ${\text{m}}^2$ square mein merge ho raha hai, banner "$=1\text{J}$".

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Step 7 — Watt: energy per second

KYA HAI. Power yeh hai ki energy kitni tezi se deliver hoti hai — energy divided by time. Joule ko ek time brick se divide karo.

$$P=\frac{W}{t}\Rightarrow [P]=\frac{[W]}{[t]}=\frac{{\text{kg⋅m}}^2{\text{⋅s}}^{-2}}{\text{s}}={\text{kg⋅m}}^2{\text{⋅s}}^{-3}\equiv 1\text{W}$$

Term by term:

• $W$ = energy → ${\text{kg⋅m}}^2{\text{⋅s}}^{-2}$ (Step 6).
• $t$ se divide karna → ek aur ${\text{s}}^{-1}$ add hota hai: exponent $-2$ se $-3$ ho jaata hai.

KYUN sirf time exponent move karta hai. Same rule jaise Step 4 mein ulta: ek time-division sirf time pile ko feed karta hai. Toh ${\text{s}}^{-2}\to {\text{s}}^{-3}$ jabki $\text{kg}$ aur ${\text{m}}^2$ wahi rehte hain.

PICTURE. Joule block ek single orange time-divider se guzar raha hai, uska time-exponent counter $-2$ se $-3$ tick kar raha hai, banner "$=1\text{W}$".

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Step 8 — Degenerate aur edge cases (dimensionless aur "per something")

KYA HAI. Har quantity ek naya brick nahi ugaati. Do special cases jo har reader milega:

1. Dimensionless quantities — pure numbers jisme koi bhi brick nahi hota.
2. "Per something" units — jaise frequency, jisme length aur mass piles empty hain.

$$\text{(a)}\left[\frac{1}{2}\right]=1(\text{no units at all})\text{(b)}[\text{frequency}]=\frac{1}{\text{s}}={\text{s}}^{-1}\equiv 1\text{Hz}$$

• $\frac{1}{2}$, $\pi$, radians mein angles, refractive index: saare bricks cancel ho jaate hain → dimensionless. Isliye parent note ke Example 2 mein $\frac{1}{2}m{v}^2$ ka $\frac{1}{2}$ ignore kiya ja sakta tha.
• Frequency $=1/\text{period}$: mass pile aur length pile empty hain; sirf ek single time-division rehta hai. Ek perfectly valid unit jo zero metre aur zero kilogram bricks se bana hai.

KYUN yeh dikhate hain. Ek recipe machine ko empty recipe bhi handle karni chahiye. Agar kabhi saare bricks cancel ho jaayein, yeh error nahi hai — iska matlab hai quantity ek pure number hai. Aur $\text{Hz}={\text{s}}^{-1}$ jaisa unit prove karta hai ki ek "derived" unit sirf saat bricks mein se ek use kar sakta hai.

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Ek-picture summary

Ek flow, teen starting bricks, chaar snaps:

$$\underset{\text{base bricks}}{\underbrace{\text{m},\text{kg},\text{s}}}\underset{÷t}{\overset{\text{÷ time}}{\to }}{\text{m⋅s}}^{-1}\underset{÷t}{\overset{\text{÷ time}}{\to }}{\text{m⋅s}}^{-2}\underset{\times m}{\overset{\text{× mass}}{\to }}\text{N}\underset{\times d}{\overset{\text{× length}}{\to }}\text{J}\underset{÷t}{\overset{\text{÷ time}}{\to }}\text{W}$$

Recall Feynman retelling plain words mein

Maine teen toy bricks se shuru kiya: ek ruler (metre), ek weight (kilogram), aur ek stopwatch (second). Pehle maine poochha "kitna fast?" aur ruler ko stopwatch se divide kiya → woh speed hai, $\text{m/s}$. Phir maine poochha "speed kitni tezi se change ho rahi hai?" aur dobara stopwatch se divide kiya → acceleration, ${\text{m/s}}^2$. Phir Newton ne whisper kiya "push = weight × acceleration", toh maine weight brick click kiya aur mila newton. Phir maine kuch room mein push kiya — force times distance — toh maine ek aur ruler click kiya, aur single metre ban gaya metre-squared: woh hai joule, energy. Phir maine poochha "energy kitni tezi se spend ho rahi hai?" — ek baar aur stopwatch se divide kiya — aur mila watt, power. Raaste mein maine do odd fellows notice kiye: pure numbers jaise $\frac{1}{2}$ jo koi brick nahi carry karte, aur frequency jo sirf stopwatch carry karta hai. Punchline: maine ek baar bhi koi naya brick invent nahi kiya. Har fancy unit bas wohi teen toys the jo ek pattern mein snap kiye the jo physics equation ne mujhe use karne ko kaha.

Recall

Newton banane mein kaun se base bricks lagte hain? ::: ${\text{kg⋅m⋅s}}^{-2}$ — mass × length ÷ time². Joule ko watt mein convert karne ke liye kaun sa operation hota hai? ::: Time se divide karna (× ${\text{s}}^{-1}$), toh ${\text{s}}^{-2}\to {\text{s}}^{-3}$. Force derived kyun hai, base kyun nahi? ::: Yeh $F=ma$ se nikalta hai; koi alag standard exist nahi karta. Dimensionless quantity ka unit kya hota hai? ::: Kuch nahi — saare bricks cancel ho jaate hain (pure number). Frequency saat bricks mein se kitne use karta hai? ::: Ek — sirf second, ${\text{s}}^{-1}$.