Physical quantities — fundamental and derived: Question bank

1.1.1 · D5 · Physics › Measurement, Vectors & Kinematics › Physical quantities — fundamental and derived

Shuru karne se pehle, do load-bearing ideas yaad karo:

Ek physical quantity hamesha number × unit hoti hai; product $n u$ hi real hai, isliye $n_{1} u_{1} = n_{2} u_{2}$ .
Fundamental quantities woh saat independent starter blocks hain (m, kg, s, A, K, mol, cd); derived quantities woh sab hain jo inhe multiply, divide, aur powers lekar banate hain.

Units algebra ki tarah kyun behave karte hain — dekho, faith pe mat lo. Ek rectangle jiska width $3 m$ aur height $2 m$ hai, uska area tum unit squares mein count karte ho: har square $1 m \times 1 m = 1 m^{2}$ ka hai. Numbers multiply karna ( $3 \times 2 = 6$ ) aur units multiply karna ( $m \times m = m^{2}$ ) dono tiling ka same physical act hain. Yahi poora reason hai ki units ordinary algebra ki tarah multiply aur divide karte hain.

Figure — Physical quantities — fundamental and derived

True or false — justify

Har item neeche ek statement hai. Decide karo true or false, lekin tera answer mein reason hona chahiye, sirf verdict nahi.

Angle item se pehle neeche radian figure dekho. Radians mein angle literally arc length divided by radius hai — do lengths, ek doosre ke upar — toh metres cancel ho jaate hain aur kuch nahi bachta sirf ek pure number.

"Weight aur mass ek hi physical quantity hain."

False. Mass (kg) ek fundamental quantity hai — matter ki amount, jo kahin bhi unchanged rehti hai. Weight force

W = m g

(newtons) hai, ek derived quantity jo Moon pe kam ho jaati hai jahan

g

smaller hota hai.

"Radians mein angle ek fundamental quantity hai kyunki tum ise protractor se measure karte ho."

False. Radian arc length ÷ radius = length/length hai, isliye units cancel ho jaate hain (upar figure dekho): yeh dimensionless hai. Measurable hona kisi cheez ko fundamental nahi banata.

"Celsius temperature ki SI base unit hai."

False. Kelvin SI base unit hai; Celsius offset hai aur true zero se shuru nahi hoti, isliye Celsius readings ke ratios meaningless hain. Celsius ek derived, shifted scale hai.

"Do quantities jinke same base units hain woh zaroor same physical quantity hain."

False. Torque aur energy dono

kg m^{2} s^{- 2}

hain, phir bhi torque ek pseudovector hai (ek axial quantity

τ = r \times F

jiska direction mirror reflection ke under flip ho jaata hai) jabki energy ek plain scalar hai. Matching units sameness ke liye necessary hai lekin sufficient nahi.

"Agar tum unit ko chota banao, toh same quantity describe karne wala number bada ho jaata hai."

True. Kyunki

n u

invariant hai,

2 m = 200 cm

: unit ko chhota karne se same physical length describe karne ke liye ek bada count zaroori ho jaata hai.

"Physics ko zyada complete banane ke liye 'speed' ko aathwein SI base quantity ke roop mein add kar sakte hain."

False. Speed already length ÷ time hai, isliye ise base unit banana same cheez ki do conflicting definitions create karta hai. Base set minimal aur independent hona chahiye — redundancy consistency ko tod deti hai.

"Har derived unit ko saat base units ki powers ka product likh sakte hain."

True. Yahi precisely 'derived' ka matlab hai; e.g.

J = kg m^{2} s^{- 2}

Pa = kg m^{- 1} s^{- 2}

"Ek pure number jaise do lengths ka ratio ek physical quantity hai jiske paas unit hai."

False. Like quantities ka ratio dimensionless hota hai — iske paas koi unit nahi hoti. Yeh ek number hai, 'number × unit' nahi.

Spot the error

Har line mein ek flawed statement ya step hai. Mistake identify karo aur use correct karo.

Neeche dimensional grid tera check hai. Har derived unit $[kg]$ , $[m]$ , $[s]$ pe exponents ki ek row hai. Quantities multiply karne se columns add hote hain; divide karne se subtract hote hain. Grid se ek row padhlo aur neeche ki errors obvious ho jaati hain.

"1 N = kg m s⁻¹ kyunki F = ma aur a length over time hai."

Error: acceleration velocity ÷ time =

(L / T) / T = L / T^{2}

hai, isliye

[a] = m s^{- 2}

, jo deta hai

1 N = kg m s^{- 2}

. Time pe exponent

- 2

hai,

- 1

nahi.

"72 km/h convert karne ke liye main likhta hoon 72 × 1000 × 3600 m/s."

Error: hours ko seconds mein divide karna hota hai, isliye

\frac{1 h}{3600 s}

se multiply karo (3600 se divide karo), jo deta hai

72 \times \frac{1000}{3600} = 20 m s^{- 1}

, koi bada number nahi.

"Pressure ke units m/m² = m hain, isliye yeh basically ek length hai."

Error: tum poori force unit ko area se divide karte ho:

\frac{kg m s ^{- 2}}{m ^{2}} = kg m^{- 1} s^{- 2}

. Sirf length exponents

1 - 2 = - 1

combine karte hain; mass aur time units rehte hain.

"Kyunki 5 m aur 5 kg dono '5' hain, woh equal amounts measure karte hain."

Error: number apni unit ke bina meaningless hai. '5 m' aur '5 kg' bilkul alag standards se compare karte hain, isliye physical quantities comparable hi nahi hain.

"Energy aur force ke same units hain kyunki dono mein force involved hai."

Error: work = force × distance, isliye energy

kg m^{2} s^{- 2}

hai jabki force

kg m s^{- 2}

hai — woh length ke ek factor (ek metre) se differ karte hain.

"Amount of substance (mole) bas ek bada number hai, isliye yeh real base quantity nahi hai."

Error: mole saat SI base quantities mein se ek hai. Yeh elementary entities count karta hai aur genuinely mass, length aur time se independent hai.

Why questions

"Ek physical quantity ko number aur unit dono ki zaroorat kyun hoti hai?"

Unit batata hai kaunse standard se compare kiya; number batata hai us standard ke kitne copies fit hote hain. Ek bare number seconds aur kilograms mein farq nahi bata sakta.

"Exactly saat base quantities kyun — kam nahi, zyada nahi?"

Saat minimal set hai jo independent bhi hai (koi ek doosre se build nahi hota) aur sufficient bhi (har known quantity inse reach hoti hai). Kam hone se gaps aate hain; zyada hone se redundancy aa jaati hai.

"Units ordinary algebra ki tarah multiply aur divide kyun karte hain?"

Kyunki quantities ka product ek tiling hai: area = length × width plane ko unit squares se bharta hai (opening figure dekho), isliye numbers multiply hote hain aur units bhi usi step mein multiply hote hain. Units bas fixed factors hain jo saath chal rahe hain.

"Temperature base scale absolute (kelvin, not Celsius) kyun honi chahiye?"

Taaki ratios meaningful hon: 100 K se 200 K tak double karna sach mein thermal quantity double karta hai, jabki 100 °C 50 °C se 'twice as hot' nahi hai kyunki Celsius ka zero arbitrary hai.

"Weight ek derived quantity kyun hai jab hum ise daily kilograms mein quote karte hain?"

Weight ek force hai

W = m g

, newtons mein measured. Roz ki 'weight in kg' actually hamari mass hai; hum dono ko isliye mix up karte hain kyunki Earth pe

g

roughly constant hai.

Edge cases

Candela teri aankhon se weighted hai — curve dekho. Do lamps jo same physical power (watts) radiate karte hain woh brightness mein bahut alag dikh sakte hain kyunki aankhein green ( $\approx 555 nm$ ) ke liye sabse zyada respond karti hain aur deep red ya violet ke liye bahut kam. Figure woh sensitivity curve dikhata hai; kyunki yeh ek biological weighting hai, koi pure energy-per-time candela ko replace nahi kar sakta.

"Kya ratio (10 m ÷ 5 m) mein number '2' ek physical quantity hai?"

Nahi — yeh dimensionless hai. Like-for-like divide karne se metres cancel ho jaate hain, ek pure number bachta hai jiske paas koi unit nahi aur koi fundamental character nahi.

"Jab ek derived unit mein ek base quantity power zero pe appear kare toh kya hota hai?"

Power zero kuch contribute nahi karta (woh base drop out ho jaata hai). E.g. frequency

s^{- 1}

hai — length aur mass zeroth power pe appear karte hain, isliye woh unit se gayab ho jaate hain.

"Kya koi unit fractional power carry kar sakti hai?"

Haan. Square root lena exponents ko half kar deta hai:

area = m^{2} = m

, aur pendulum ki frequency

f \propto g / ℓ

deti hai

m s^{- 2} / m = s^{- 1}

. Units pe exponents wahi arithmetic follow karte hain jo kisi bhi exponent pe hoti hai — halving bilkul legal hai.

"Kya koi quantity ek unit system mein fundamental lekin doosre mein derived ho sakti hai?"

Haan. Base set ki choice ek human convention hai. Kuch systems alag starters choose karte hain, isliye jo yahan base quantity hai woh wahan derived ho sakti hai — physics unchanged rehti hai, sirf bookkeeping badlti hai.

"Kya luminous intensity (candela) truly independent hai, ya hidden energy per time hai?"

Ise ek alag base ke roop mein rakha jaata hai kyunki yeh human eye ki sensitivity curve se weighted hai (figure dekho), jo koi purely physical energy-rate capture nahi kar sakta. Woh eye-response weighting ise SI convention mein independent banati hai.

"Agar do log alag units use karein, toh underlying quantity alag hoti hai kya?"

Nahi. Kyunki

n_{1} u_{1} = n_{2} u_{2}

, product (real quantity) invariant hai. Sirf label pair (number, unit) badlta hai; nature same rehti hai.

Connections

Dimensional Analysis — upar har 'spot the error' exponent grid padhke aur bhi jaldi pakad mein aata hai.
SI Units and Prefixes — invariance $n_{1} u_{1} = n_{2} u_{2}$ in saari conversions ko power deta hai.
Newton's Second Law — error items mein use hue newton ki derivation ka source.
Errors and Significant Figures — real measured quantities uncertainty bhi carry karti hain.
Vectors — force ek vector hai aur torque ek pseudovector, ye differences sirf units se pare hain.