Worked examples — SI units — seven base units and all derived units

1.1.2 · D3 · Physics › Measurement, Vectors & Kinematics › SI units — saat base units aur saare derived units

Parent par wapas link: SI units — seven base units and all derived units. Yeh kaam Dimensional Analysis (equations ko units se check karna), Significant Figures and Errors (answers round karna), Kinematics — velocity and acceleration, Newton's Laws of Motion aur Work, Energy and Power (woh equations jo units ko define karti hain) par lean karta hai.

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The scenario matrix

Kuch bhi work karne se pehle, yahan case-classes ka poora map hai jo ek "unit derivation" question tumpar throw kar sakta hai. Neeche har worked example un cell(s) ke saath tagged hai jinhein woh cover karta hai.

Cell	Case class	Kya cheez tricky banati hai
A	Seedha forward-derivation ($F=ma$ style)	bas base units substitute karo
B	Rearrange-first (unknown is buried)	isolate the target, THEN substitute
C	Negative / fractional powers appear	units divide karne se exponents kam hote hain; signs dekho
D	Dimensionless factor present ($\frac{1}{2}$, $2\pi$, $\sin \theta$)	numbers aur pure ratios koi unit carry NAHI karte
E	Degenerate / zero input	kya hota hai jab koi quantity $0$ ho ya unit "cancels out" ho jaaye?
F	Prefix trap (km, cm, g, hour)	pehle base units mein convert karo
G	Real-world word problem	equation ko ek story se nikalo
H	Exam twist (verify a claimed formula)	kya yeh dimensionally legal bhi hai?
I	Limiting / consistency check	kya do routes ek hi base form dete hain?

Woh rule jo har cell mein bachta hai:

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Example 1 — Cell A (seedha derivation)

Forecast: aage padhne se pehle second ka exponent guess karo — kya yeh $-2$ hai ya $-3$?

1. Recipe likho: $[P]=\frac{[W]}{[t]}=\frac{{\text{kg⋅m}}^2{\text{⋅s}}^{-2}}{\text{s}}$. Yeh step kyun? Power ki defining equation energy over time hai; unit usi division ki hai.
2. Ek second se divide karo → $\text{s}$ ke exponent se $1$ ghataao: $-2-1=-3$. Yeh step kyun? Unit algebra: ${\text{s}}^1$ se divide karne par $\text{s}$ ki power ek se kam ho jaati hai.
3. Result: $[P]={\text{kg⋅m}}^2{\text{⋅s}}^{-3}\equiv 1\text{W}$.

Verify: $P=Fv$ se cross-check karo: $[F][v]=({\text{kg⋅m⋅s}}^{-2})({\text{m⋅s}}^{-1})={\text{kg⋅m}}^2{\text{⋅s}}^{-3}$. ✔ Same answer, alag equation — yeh ek consistency win hai (Cell I).

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Example 2 — Cell B + C (rearrange karo, negative powers)

Forecast: capacitance charge divided by voltage hai — kya $\text{kg}$ positive end hoga ya negative?

1. Jaane-maane base forms substitute karo: $[Q]=\text{A⋅s}$ aur $[V]={\text{kg⋅m}}^2{\text{⋅s}}^{-3}{\text{⋅A}}^{-1}$. Yeh step kyun? Yeh hum parent note mein derive kar chuke hain; reuse karo, dobara derive mat karo.
2. Divide karo: $[C]=\frac{\text{A⋅s}}{{\text{kg⋅m}}^2{\text{⋅s}}^{-3}{\text{⋅A}}^{-1}}$. Yeh step kyun? Defining equation ek division hai, isliye unit bhi ek division hai.
3. Base by base exponents subtract karo: =\text{A}^{2}\,\text{s}^{4}\,\text{kg}^{-1}\,\text{m}^{-2}.$$ **Yeh step kyun?** Kisi quantity se divide karne par *uske* exponents ka sign flip ho jaata hai, phir add karte hain. $\text{kg}$ denominator mein upar tha, isliye woh $\text{kg}^{-1}$ ban jaata hai — ek **negative power**, exactly Cell C.
4. Result: $[C]={\text{A}}^2{\text{⋅s}}^4{\text{⋅kg}}^{-1}{\text{⋅m}}^{-2}\equiv 1\text{F}$.

Verify: parent note ki reference table se match karta hai. ✔

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Example 3 — Cell D (dimensionless factors)

Forecast: teeno energy jaisi lagti hain. Koi impostor toh nahi?

1. (a) $[\frac{1}{2}m{v}^2]$: $\frac{1}{2}$ drop karo (pure number, Cell D), phir $\text{kg}\cdot ({\text{m⋅s}}^{-1}{)}^2={\text{kg⋅m}}^2{\text{⋅s}}^{-2}=\text{J}$. ✔ Yeh step kyun? Ek dimensionless coefficient kabhi bhi unit nahi badal sakta; sirf variables count karte hain.
2. (b) $[2\pi mgh]$: $2\pi$ drop karo (yeh bhi dimensionless hai), phir $\text{kg}\cdot ({\text{m⋅s}}^{-2})\cdot \text{m}={\text{kg⋅m}}^2{\text{⋅s}}^{-2}=\text{J}$. ✔ Yeh step kyun? $g$ ek acceleration ${\text{m⋅s}}^{-2}$ hai; $2\pi$ sirf geometry hai, koi unit nahi.
3. (c) $[p^2/(2m)]$: $\frac{({\text{kg⋅m⋅s}}^{-1}{)}^2}{\text{kg}}=\frac{{\text{kg}}^2{\text{⋅m}}^2{\text{⋅s}}^{-2}}{\text{kg}}={\text{kg⋅m}}^2{\text{⋅s}}^{-2}=\text{J}$. ✔ Yeh step kyun? Momentum unit ko square karo (har exponent double ho jaata hai), phir mass se divide karo (subtract karo).

Verify: teeno ${\text{kg⋅m}}^2{\text{⋅s}}^{-2}$ par collapse karti hain. Teeno genuine energies hain. ✔

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Example 4 — Cell F + G (word problem mein prefix trap)

Forecast: grams aur cm use karne se number $1000\times$ off hoga ya $100\times$ — ya dono?

1. Pehle base units mein convert karo. Mass already $1200\text{kg}$ hai (grams SI equation mein mat daalo). Height $45\text{m}$ ($4500\text{cm}$ use mat karo). Time $30\text{s}$. Yeh step kyun? Equation $mgh/t$ watts produce karta hai sirf tabhi jab har input base SI mein ho. Grams aur centimetres tumhara answer silently ten ki galat powers se multiply kar denge (Cell F).
2. Substitute karo: $P=\frac{(1200)(9.8)(45)}{30}\text{W}$. Yeh step kyun? Clean base units ke saath number seedha watts mein aata hai — end mein koi extra conversion nahi chahiye.
3. Numerator compute karo: $1200\times 9.8=11760$; $\times 45=529200$. $30$ se divide karo: $P=17640\text{W}=17.6\text{kW}.$ Yeh step kyun? 3 significant figures tak round kiya (Significant Figures and Errors).

Verify: unit check $\frac{\text{kg}\cdot {\text{m⋅s}}^{-2}\cdot \text{m}}{\text{s}}={\text{kg⋅m}}^2{\text{⋅s}}^{-3}=\text{W}$. ✔ Aur $17640\text{W}$ ≈ 24 horsepower — hill par car ke liye believable. ✔

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Example 5 — Cell E (degenerate / zero input)

Forecast: agar force phir bhi lag raha hai, toh kya koi kaam ho raha hai? Kya unit tab bhi exist karti hai jab $d=0$ ho?

1. Work hai $W=Fd$, isliye $[W]=\text{N⋅m}={\text{kg⋅m}}^2{\text{⋅s}}^{-2}=\text{J}$ — unit badli nahi. Yeh step kyun? Unit ek type hai; yeh sirf isliye gayab nahi ho jaati kyunki ek value zero hai. $0\text{J}$ bhi joules mein hi measure hota hai.
2. Value: $W=F\times 0=0\text{J}$. Yeh step kyun? Cell E — ek degenerate (zero) input number ko collapse karta hai lekin dimension ko nahi. Zero displacement ⇒ zero work, chahe kitni bhi force lagao.
3. $W=Fd\cos \theta$ se $\theta =90^{\circ }$ par compare karo: $\cos 90^{\circ }=0$, isliye phir bhi $W=0\text{J}$ motion ke bawajood — kyunki force path ke perpendicular hai. Yeh step kyun? $\cos \theta$ dimensionless hai (Cell D), isliye woh value ko zero kar deta hai lekin unit kabhi nahi badalta.

Verify: $F\cdot 0=0$ aur $\cos 90^{\circ }=0$ ⇒ $W=0\text{J}$ dono tarike se. ✔

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Example 6 — Cell H (exam twist: kya formula legal bhi hai?)

Forecast: inmen se ek seconds deta hai; ek kuch absurd deta hai. Kaun?

1. Original handle karo. Root ke andar: $\frac{L}{g}=\frac{\text{m}}{{\text{m⋅s}}^{-2}}={\text{s}}^2$. Yeh step kyun? Metre ko (metre per second²) se divide karo: metres cancel ho jaate hain aur denominator mein ${\text{s}}^{-2}$ flip hokar ${\text{s}}^{+2}$ ban jaata hai.
2. Square root lo: $\sqrt{{\text{s}}^2}=\text{s}$. Dimensionless $2\pi$ se multiply karo: phir bhi $\text{s}$. Yeh step kyun? Square root har exponent ko half kar deta hai ($2\to 1$); $2\pi$ koi unit carry nahi karta (Cell D). Result: seconds — ek valid period. ✔
3. Rival handle karo. Andar: $\frac{L}{g^2}=\frac{\text{m}}{({\text{m⋅s}}^{-2}{)}^2}=\frac{\text{m}}{{\text{m}}^2{\text{⋅s}}^{-4}}={\text{m}}^{-1}{\text{⋅s}}^4$. Yeh step kyun? $g$ ko square karne par uske exponents double ho jaate hain; phir $\text{m}$ mein divide karo.
4. Square root: $\sqrt{{\text{m}}^{-1}{\text{⋅s}}^4}={\text{m}}^{-1/2}{\text{⋅s}}^2$. Yeh step kyun? Exponents half karne par metre par $-\frac{1}{2}$ aur second par $2$ aata hai — yeh ek time nahi hai. Rival formula dimensionally impossible hai. ✗

Verify: original → $\text{s}$; rival → ${\text{m}}^{-1/2}{\text{⋅s}}^2$. Sirf original ek valid period hai. ✔ (Figure dekho — do "unit towers" alag heights par end hoti hain.)

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Example 7 — Cell C + I (fractional powers, do routes agree karte hain)

Forecast: ek square root of (force ÷ linear density) — kya yeh sach mein speed ke roop mein aata hai?

1. Andar ki units: $[T_{\text{en}}]={\text{kg⋅m⋅s}}^{-2}$, $[\mu ]={\text{kg⋅m}}^{-1}$. Yeh step kyun? Tension ek force hai; $\mu$ kilograms per metre hai.
2. Divide karo: $\frac{{\text{kg⋅m⋅s}}^{-2}}{{\text{kg⋅m}}^{-1}}={\text{m}}^{1-(-1)}{\text{⋅s}}^{-2}={\text{m}}^2{\text{⋅s}}^{-2}$. Yeh step kyun? $\text{kg}$ cancel ho jaata hai ($1-1=0$); metre exponent $1-(-1)=2$; ${\text{s}}^{-2}$ untouched rehta hai.
3. Square root exponents half kar deta hai: $\sqrt{{\text{m}}^2{\text{⋅s}}^{-2}}={\text{m⋅s}}^{-1}$. Yeh step kyun? $\frac{2}{2}=1$ aur $\frac{-2}{2}=-1$ — yeh ek genuine speed hai (Cell C fractional step whole powers par land karta hai).

Verify: ${\text{m⋅s}}^{-1}$ exactly velocity ki unit hai Kinematics — velocity and acceleration se. ✔ Example 1 ke cross-checks ke saath consistency confirm karta hai ki humari unit-algebra rules har jagah ek hi tarah apply ho rahi hain (Cell I).

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Example 8 — Cell B + G (word problem, hidden rearrange)

Forecast: 2 minutes mein ek prefix trap chhupi hai — compute karne se pehle pakdo.

1. Time ko base units mein convert karo: $2\text{min}=120\text{s}$ (Cell F — SI power calculation mein kabhi minutes mat chhodna). Yeh step kyun? Watt joules per second hai; "2" feed karna $60\times$ se over-count karega.
2. $P=\frac{W}{t}=\frac{360000}{120}\text{W}$ lagao. Yeh step kyun? Power energy per unit time hai — defining equation.
3. Compute karo: $P=3000\text{W}=3\text{kW}$. Yeh step kyun? 3 kW ka kettle ek realistic value hai — ek sanity anchor (Cell G).

Verify: unit chain $\frac{\text{J}}{\text{s}}=\frac{{\text{kg⋅m}}^2{\text{⋅s}}^{-2}}{\text{s}}={\text{kg⋅m}}^2{\text{⋅s}}^{-3}=\text{W}$. ✔ Number: $360000/120=3000$. ✔

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Recap

Recall Which cell is which?

Seedha substitution = Cell ::: A Target isolate karne ke liye pehle rearrange karo = Cell ::: B Negative ya fractional exponents aate hain = Cell ::: C Unit check mein $\frac{1}{2}$, $2\pi$, $\cos \theta$ ignore karo = Cell ::: D Zero displacement $0\text{J}$ deta hai lekin unit joules rehti hai = Cell ::: E Convert km/cm/g/minutes to base units before substituting = Cell ::: F Farad ki base-unit form? ::: ${\text{A}}^2{\text{⋅s}}^4{\text{⋅kg}}^{-1}{\text{⋅m}}^{-2}$ Ek pendulum period $2\pi \sqrt{L/g}$ ki units kya hain? ::: seconds ($\text{s}$) Wave speed $\sqrt{T_{\text{en}}/\mu }$ ki units kya hain? ::: ${\text{m⋅s}}^{-1}$ Kettle: $360000\text{J}$ in $2\text{min}$ → power? ::: $3000\text{W}$ ($3\text{kW}$)