Dimensional analysis — checking equations, deriving relations: Question bank

1.1.3 · D5 · Physics › Measurement, Vectors & Kinematics › Dimensional analysis — checking equations, deriving relation

True or false — justify

Ek dimensionally consistent equation guaranteed physically correct hoti hai.

False. Consistency zaroori hai par kaafi nahi — yeh

s = u t + a t^{2}

mein missing

\frac{1}{2}

nahi dekh sakti, na hi

E = m c^{2}

aur

E = 3 m c^{2}

mein fark bata sakti hai. Pass karna green light hai, proof nahi.

Agar koi equation dimensional check fail kar de, toh bhi woh sahi ho sakti hai.

False. Failed check decisive hota hai: agar do added terms ke dimensions alag hain toh equation definitely galat hai, koi exception nahi.

Quantities

km/h

aur

m/s

ke alag dimensions hote hain.

False. Ye sirf unit mein alag hain (chosen size); dono speeds hain aur inka dimension same hai

[L T^{- 1}]

. Dimension "kind" hota hai, unit "size" hoti hai.

Radians mein angle ka dimension

[L]

hota hai kyunki yeh arc length over radius hai.

False. Yeh length ÷ length hai, toh

L

cancel ho jaate hain: angle dimensionless hota hai

[M^{0} L^{0} T^{0}]

. Isliye hi

sin θ

allowed hai.

1 0^{c}

mein exponent

c

dimensionless hona chahiye.

True. Exponent ek pure number hota hai; kisi base ko "2 metres" ki power tak uthana meaningless hai, isliye har exponent (jaise har angle aur log argument) ka dimension

[M^{0} L^{0} T^{0}]

hota hai.

Dimensional analysis pendulum period mein numerical factor

2 π

determine kar sakta hai.

False. Dimensionless constants is method ke liye invisible hote hain — yeh sirf

T \propto ℓ / g

deta hai, lekin

k = 2 π

full theory ya experiment se aana chahiye.

Ek correct equation ke dono sides hamesha same dimension share karte hain.

True. Equate karne ka matlab hai "dono sides par same kind ki cheez," isliye homogeneity force karta hai ki

[LHS] = [RHS]

ho.

Equation

s = u t + \frac{1}{2} a t^{2}

ko dimensional analysis se derive kiya ja sakta hai.

False. Yeh do terms ka sum hai, aur method sirf powers ke single products handle karta hai — yeh multi-term formulas nahi bana sakta.

Same dimension wali do quantities ek hi physical quantity hoti hain.

False. Torque aur energy dono

[M L^{2} T^{- 2}]

rakhte hain, phir bhi ek turning effect hai aur doosra kaam karne ki capacity — same dimension, alag physics.

Spot the error

v = u + a t^{2}

— yeh theek hai, dono sides velocities hain."

Galat.

[a t^{2}] = (L T^{- 2}) (T^{2}) = L

, yeh ek length hai, velocity

L T^{- 1}

nahi. Yeh term

u

aur

v

se clash karta hai; sahi law hai

v = u + a t

F = \frac{m v ^{2}}{r ^{2}}

circular motion ke liye — dimensionally ek force hai."

Galat.

[m v^{2} / r^{2}] = M (L^{2} T^{- 2}) / L^{2} = M T^{- 2}

, jo

M L T^{- 2}

nahi hai. Sahi form

m v^{2} / r

hai (

r

ki ek power), jo ek genuine force deta hai.

θ = ω t + \frac{1}{2} α

— angle barabar angular speed times time plus half angular acceleration."

Galat.

[α] = T^{- 2}

angle ki tarah dimensionless nahi hai. Missing

t^{2}

(

\frac{1}{2} α t^{2}

)

T^{- 2} \cdot T^{2} =

dimensionless restore karta hai.

E = \frac{1}{2} m v

meri quick glance mein pass ho gaya, energy mass times speed hoti hai."

Galat.

[m v] = M L T^{- 1}

ek momentum hai, energy

M L^{2} T^{- 2}

nahi. Kinetic energy ko

v^{2}

chahiye:

\frac{1}{2} m v^{2}

y = A sin (k x - ω t)

mein,

k

koi bhi dimension rakh sakta hai."

Galat. Argument

k x - ω t

dimensionless hona chahiye, isliye har piece bhi:

[k x]

dimensionless

\Rightarrow [k] = L^{- 1}

, aur

[ω t]

dimensionless

\Rightarrow [ω] = T^{- 1}

"Kyunki

lo g

koi bhi positive number le sakta hai,

lo g (x)

jahan

x

metres mein hai, theek hai."

Galat. Log ka argument dimensionless hona chahiye —

lo g

x

ki powers mein expand hota hai, jinhe tab tak add nahi kiya ja sakta jab tak

x

pure number na ho. Physicists

lo g (x / x_{0})

likhte hain ise fix karne ke liye.

Why questions

sin

cos

e^{x}

, aur

ln

ke arguments dimensionless kyun hone chahiye?

Ye functions infinite sums ke barabar hote hain jaise

x - \frac{x ^{3}}{6} + \dots

;

x

x^{3}

mein add karna tabhi legal hai jab

x

pure number ho, isliye poora argument dimensionless hona chahiye.

Homogeneity "5 metres + 3 seconds" add karna kyun forbid karta hai?

Addition do same kind ki amounts ko merge karta hai; koi aisa number nahi hai jo ek saath length aur time ho, isliye is sum ka koi matlab nahi — nature kabhi aisa term nahi likhti.

Dimensional analysis kaise reveal karta hai ki pendulum ka period mass se independent hai?

M

ki power match karne se

b = 0

aata hai, kyunki na

ℓ

(ek length) mein, na

g

(ek acceleration) mein koi mass hai — isliye unke kisi combination mein answer mein mass aa hi nahi sakti.

Ek passed dimensional check ek sign error kyun hide kar sakta hai jaise

v^{2} = u^{2} - 2 a s

+ 2 a s

mein?

Dono terms

u^{2}

aur

2 a s

ka dimension

L^{2} T^{- 2}

share karte hain chahe kaun sa bhi sign join kare; dimensions kind describe karte hain, same-kind terms ke beech arithmetic

+

-

nahi.

Power-matching trick mechanics mein exactly teen equations kyun deta hai?

Kyunki

M

L

, aur

T

teen independent "axes" hain, aur homogeneity demand karta hai ki powers har axis par alag-alag match karein — ek equation per base dimension.

Dimensional analysis tab kyun fail ho sakta hai jab koi quantity chaar variables par depend kare?

Chaar unknown exponents hain lekin sirf teen matching equations (

M, L, T

) — system underdetermined ho jaata hai — ek exponent free rehta hai, isliye form uniquely pin down nahi hoti.

Pure number

2

[2 a s] = L^{2} T^{- 2}

mein drop karna kyun justified hai?

Pure number ka dimension

[M^{0} L^{0} T^{0}]

hota hai, isliye isse multiply karne se dimensionally kuch nahi badalta — "

2

" simply method ko dikhta hi nahi.

Edge cases

Kisi dimensionless quantity ka "dimension" kuch nahi likhte ya

M^{0} L^{0} T^{0}

likhte hain?

Dono ka matlab same hai — ek pure number har base ko power zero par rakhta hai,

[M^{0} L^{0} T^{0}]

, jise hum aksar bas "dimensionless" kehte hain.

Kya do dimensionally correct equations ek saath dono galat ho sakti hain?

Haan.

s = u t + a t^{2}

aur

s = u t + 3 a t^{2}

dono consistent hain phir bhi dono galat hain (sirf

\frac{1}{2}

sahi hai) — consistency garbage filter karta hai lekin kaafi impostors ko rakh leta hai.

Agar kisi equation ka har term dimensionless hai, toh kya check automatic hai?

Effectively haan homogeneity ke liye — sab terms already

[M^{0} L^{0} T^{0}]

share karte hain — lekin hidden numbers, signs, ya functional form ke baare mein phir bhi kuch pata nahi chalta.

Agar solve kiye gaye exponents fractions mein aayein, jaise

c = - \frac{1}{2}

, toh iska matlab kya hai?

Kuch galat nahi hai — fractional powers legal hain (yeh ek square root signal karte hain, jaise

ℓ / g

mein); dimensions sirf yeh require karte hain ki powers balance karein, whole numbers hों yeh zarоori nahi.

Kya dimensional analysis kisi aisi equation par apply hota hai jo dimensionally correct hai lekin ek extra unit hai jaise unmatched angle offset?

Yeh pakad nahi sakta — angle offset (jaise

sin (ω t + ϕ)

galat

ϕ

ke saath) dimensionless hai, isliye check pass ho jaata hai jabki physics phir bhi galat ho sakti hai.

Kya method ek purely numerical relation derive kar sakta hai jisme koi dimensions nahi (jaise ek ratio law)?

Nahi. Jab har quantity dimensionless ho toh match karne ke liye koi

M, L, T

powers hain hi nahi, isliye homogeneity zero constraints deta hai — trick ke paas kuch karne ko nahi bachta.

Connections

Dimensional analysis — checking equations, deriving relations — woh parent jise yeh bank drill karta hai.
Equations of motion (kinematics) — kaafi "spot the error" cases ka source.
Units and the SI system — jahan dimension-vs-unit trap aata hai.
Buckingham Pi theorem — "unknowns vs equations" limit ko formally describe karta hai.
Vectors — components and addition — same-kind-adds-to-same-kind, geometrically.