Dimension of a Physical Quantity

Q 1 :

The equation of stationary wave is: [2024]

$y = 2 a \sin (\frac{2 πnt}{λ}) \cos (\frac{2 πx}{λ}) .$

Which of the following is not correct?

The dimensions of $x$ is $[L]$
The dimensions of $n t$ is $[L]$
The dimensions of $n$ is $[L T^{- 1}]$
The dimensions of $n / l$ is $[T]$

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(4)

$y = 2 a \sin (\frac{2 πnt}{λ}) \cos (\frac{2 πx}{λ})$

(i) $[x] = L$

(ii) $[\frac{2 πnt}{λ}] = 1 \Rightarrow [n t] = 1 \cdot [λ] = L$

(iii) $[n t] = L \Rightarrow [n] = L T^{- 1}$

(iv) $[\frac{n}{λ}] = \frac{L T^{- 1}}{L} = T^{- 1}$

Q 2 :

Applying the principle of homogeneity of dimensions, determine which one is correct, where T is time period, G is gravitational constant, M is mass, r is radius of orbit. [2024]

$T^{2} = \frac{4 π^{2} r^{3}}{G M}$
$T^{2} = \frac{4 π^{2} r^{2}}{G M}$
$T^{2} = 4 π^{2} r^{3}$
$T^{2} = \frac{4 π^{2} r}{G M^{2}}$

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(1)

According to principle of homogeneity dimension of LHS should be equal to dimensions of RHS. $[T^{2}] = T^{2},$

Dimension of G is $[M^{- 1} L^{3} T^{- 2}]$

Now,

(1) $[\frac{4 π^{2} r^{3}}{G M}] = \frac{L^{3}}{M^{- 1} L^{3} T^{- 2} M^{1}} = T^{2}$

(2) $[\frac{4 π^{2} r^{2}}{G M}] = \frac{L^{2}}{M^{- 1} L^{3} T^{- 2} M^{1}} = L^{- 1} T^{2}$

(3) $[4 π^{2} r^{3}] = L^{3}$

(4) $[\frac{4 π^{2} r}{G M^{2}}] = \frac{L}{M^{- 1} L^{3} T^{- 2} M^{2}} = M^{- 1} L^{- 2} T^{2}$

Q 3 :

If $G$ be the gravitational constant and $u$ be the energy density then which of the following quantity have the dimensions as that of the $\sqrt{u G} .$ [2024]

Force per unit mass
Gravitational potential
pressure gradient per unit mass
Energy per unit mass

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(1)

$[u G] = [(M^{1} L^{- 1} T^{- 2}) (M^{- 1} L^{3} T^{- 2})]$

$[u G] = [M^{0} L^{2} T^{- 4}]$

$[\sqrt{u G}] = [L^{1} T^{- 2}]$

Q 4 :

What is the dimensional formula of $a b^{- 1}$ in the equation $(P + \frac{a}{V^{2}}) (V - b) = R T,$ where letters have their usual meaning. [2024]

$[M^{6} L^{7} T^{4}]$
$[M L^{2} T^{- 2}]$
$[M^{- 1} L^{5} T^{3}]$
$[M^{0} L^{3} T^{- 2}]$

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(2)

$[V] = [b]$ It means the dimension of $b = [L^{3}]$

$[P] = [\frac{a}{V^{2}}] \Rightarrow [a] = [P V^{2}] = [M L^{- 1} T^{- 2}] [L^{6}]$

Dimension of $a = [M L^{5} T^{- 2}]$

$∴ a b^{- 1} = \frac{[M L^{5} T^{- 2}]}{[L^{3}]} = [M L^{2} T^{- 2}]$

Q 5 :

Match List-I with List-II: [2024]

List I List II

A Torque (I) $[M^{1} L^{2} T^{- 2} A^{- 2}]$

B Magnetic Field (II) $[L^{2} A^{1}]$

C Magnetic moment (III) $[M^{1} T^{- 2} A^{- 1}]$

D Permeability of free space (IV) $[M^{1} L^{2} T^{- 2}]$

Choose the correct answer from the options given below:

A-III, B-l, C-II, D-IV
A-IV, B-II, C-III, D-I
A-IV, B-III, C-II, D-I
A-I, B-III, C-II, D-IV

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(3)

$[\vec{τ}] = [\vec{r} \times \vec{F}] = [M L^{2} T^{- 2}]$

$[B] = [\frac{M L T^{- 2}}{A T L T^{- 1}}] = [M A^{- 1} T^{- 2}]$

$[M] = [I \times A] = [A L^{2}]$

$B = \frac{μ_{0}}{4 π} \frac{I d l \sin θ}{r^{2}} \Rightarrow [μ] = [\frac{B r^{2}}{I d l}]$

$\Rightarrow [μ] = [\frac{M T^{- 2} A^{- 1} \times L^{2}}{A L}] = [M L T^{- 2} A^{- 2}]$

Q 6 :

Given below are two statements: [2024]

Statements (I): Dimensions of specific heat is $[L^{2} T^{- 2} K^{- 1}]$

Statements (Il): Dimensions of gas constant is $[M L^{2} T^{- 1} K^{- 1}]$

In the light of the above statements choose the most appropriate answer from the options given below:

Both statement (I) and statement (lI) are correct
Statement (I) is incorrect but statement (II) is correct
Statement (I) is correct but statement (II) is incorrect
Both statement (I) and statement (II) are incorrect

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(3)

Specific heat, $s = \frac{∆ Q}{m ∆ T} = \frac{[M L^{2} T^{- 2}]}{[M K]} = [L^{2} T^{- 2} K^{- 1}]$

Statement-(I) is correct

$P V = n R T \Rightarrow R = \frac{P V}{n T}$

$[R] = \frac{[M L^{- 1} T^{- 2}] [L^{3}]}{[m o l] [K]}$

$[R] = [M L^{2} T^{- 2} m o l^{- 1} K^{- 1}]$

Statement-II is incorrect

Q 7 :

If $\in_{0}$ is the permittivity of free space and E is the electric field, then $\in_{0} E^{2}$ has the dimensions [2024]

$[M^{0} L^{- 2} T A]$
$[M L^{- 1} T^{- 2}]$
$[M^{- 1} L^{- 3} T^{4} A^{2}]$
$[M L^{2} T^{- 2}]$

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(2)

Energy per unit volume = $\frac{1}{2} \in_{0} E^{2}$

$\frac{1}{2} \in_{0} E^{2} \Rightarrow [M L^{- 1} T^{- 2}] = \in_{0} E^{2}$

Q 8 :

The dimensional formula of latent heat is [2024]

$[M^{0} L T^{- 2}]$
$[M L T^{- 2}]$
$[M^{0} L^{2} T^{- 2}]$
$[M L^{2} T^{- 2}]$

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(3)

$Q = m L . \Rightarrow [L] = \frac{[Q]}{[m]}$

$[L] = [\frac{M L^{2} T^{- 2}}{M}] = [M^{0} L^{2} T^{- 2}]$

Q 9 :

The de-Broglie wavelength associated with a particle of mass $m$ and energy $E$ is $h / \sqrt{2 m E} .$ The dimensional formula for Planck's constant is [2024]

$[M L^{- 1} T^{- 2}]$
$[M L^{2} T^{- 1}]$
$[M L T^{- 2}]$
$[M^{2} L^{2} T^{- 2}]$

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(2)

Given $λ = \frac{h}{\sqrt{2 m E}}$

$\Rightarrow h = λ \sqrt{2 m E}$ or $h = λ p$

So dimensional formula of Planck's constant

$[h] = [l e n g t h] [M a s s \times v e l o c i t y]$

$= [L^{1}] [M^{1} L^{1} T^{- 1}] = [M L^{2} T^{- 1}]$

Q 10 :

Given below are two statements: [2024]

Statement (I): Planck's constant and angular momentum have same dimensions.

Statement (II): Linear momentum and moment of force have same dimensions.

In the light of the above statements, choose the correct answer from the options given below:

Statement I is true but Statement II is false
Both Statement I and Statement II are false
Both Statement I and Statement II are true
Statement I is false but Statement II is true

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(1)

$[h] = M L^{2} T^{- 1}$

$[L] = M L^{2} T^{- 1}$

$[P] = M L T^{- 1}$

$[τ] = M L^{2} T^{- 2}$

(Here $h$ is Planck's constant, $L$ is angular momentum, $P$ is linear momentum and $τ$ is moment of force)

Q 11 :

The equation of state of a real gas is given by $(P + \frac{a}{V^{2}}) (V - b) = R T,$ where $P, V$ and $T$ are pressure, volume and temperature respectively and R is the universal gas constant. The dimensions of $\frac{a}{b^{2}}$ is similar to that of: [2024]

$P V$
$P$
$R T$
$R$

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(2)

$[P] = [\frac{a}{V^{2}}] \Rightarrow [a] = [P V^{2}]$

And $[V] = [b]$

$\frac{[a]}{[b^{2}]} = \frac{[P V^{2}]}{[V^{2}]} = [P]$

Q 12 :

Match List-I with List-II [2024]

List-I List-II

A. Coefficient of viscosity I. $[M L^{2} T^{- 2}]$

B. Surface tension II. $[M L^{2} T^{- 1}]$

C. Angular momentum III. $[M L^{- 1} T^{- 1}]$

D. Rotational kinetic energy IV. $[M L^{0} T^{- 2}]$

A-II, B-I, C-IV, D-III
A-I, B-II, C-III, D-IV
A-III, B-IV, C-II, D-I
A-IV, B-III, C-II, D-I

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(3)

(A) Coefficient of viscosity

$F = 6 πηrv$

$[M^{1} L^{1} T^{- 2}] = [η] [M^{0} L^{1} T^{0}] [M^{0} L^{1} T^{- 1}]$

$[η] = \frac{[M^{1} L^{1} T^{- 2}]}{[M^{0} L^{1} T^{0}] [M^{0} L^{1} T^{- 1}]} = [M^{1} L^{- 1} T^{- 1}] \to (I I I)$

(B) Surface Tension

$(T) = \frac{F}{l} = \frac{[M^{1} L^{1} T^{- 2}]}{[M^{0} L^{1} T^{0}]} = [M^{1} L^{0} T^{- 2}] \to (I V)$

(C) Angular momentum

$m v r = [M^{1} L^{0} T^{0}] [M^{0} L^{1} T^{- 1}] [M^{0} L^{1} T^{0}]$

$= [M^{1} L^{2} T^{- 1}] \to (I I)$

(D) Rotational Kinetic Energy

$[M^{1} L^{2} T^{- 2}] \to (I)$

Q 13 :

If mass is written as $m = k c^{P} G^{- 1 / 2} h^{1 / 2}$ then the value of $P$ will be (Constants have their usual meaning with $k$ a dimensionless constant) [2024]

1/2
1/3
2
- 1/3

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(1)

$m = K C^{P} G^{- 1 / 2} h^{1 / 2}$ ...(1)

$G = \frac{F r^{2}}{m_{1} \cdot m_{2}} = \frac{M L T^{- 2} \cdot L^{2}}{M^{2}} = [M^{- 1} L^{3} T^{- 2}]$

Now using (1)

$[M] = {[L T^{- 1}]}^{P} {[M^{- 1} L^{3} T^{- 2}]}^{- 1 / 2} {[M L^{2} T^{- 1}]}^{1 / 2}$

Compare the dimension of T from L.H.S and R.H.S then

$- P + \frac{1}{2} = 0 \Rightarrow P = \frac{1}{2}$

Q 14 :

A force is represented by $F = a x^{2} + b t^{1 / 2}$ where $x$ = distance and $t$ = time. The dimensions of $b^{2} / a$ are [2024]

$[M L^{2} T^{- 3}]$
$[M L^{3} T^{- 3}]$
$[M L^{- 1} T^{- 1}]$
$[M L T^{- 2}]$

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(2)

$F = a x^{2} + b t^{1 / 2}$

$[a] = \frac{[F]}{[x^{2}]} = [M^{1} L^{- 1} T^{- 2}]$

$[b] = \frac{[F]}{[t^{1 / 2}]} = [M^{1} L^{1} T^{- 5 / 2}]$

$[\frac{b^{2}}{a}] = \frac{[M^{2} L^{2} T^{- 5}]}{[M^{1} L^{- 1} T^{- 2}]} = [M^{1} L^{3} T^{- 3}]$

Q 15 :

Consider two physical quantities A and B related to each other as $E = \frac{B - x^{2}}{A t}$ where $E, x$ and $t$ have dimensions of energy, length and time respectively. The dimension of A B is [2024]

$L^{- 2} M^{1} T^{0}$
$L^{2} M^{- 1} T^{1}$
$L^{- 2} M^{- 1} T^{1}$
$L^{0} M^{- 1} T^{1}$

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(2)

$[B] = L^{2}$

$A = \frac{x^{2}}{t E} = \frac{L^{2}}{T M L^{2} T^{- 2}} = \frac{1}{M T^{- 1}}$

$[A] = M^{- 1} T$

$[A B] = [L^{2} M^{- 1} T^{1}]$

Q 16 :

The dimensional formula of angular impulse is [2024]

$[M L^{- 2} T^{- 1}]$
$[M L^{2} T^{- 2}]$
$[M L T^{- 1}]$
$[M L^{2} T^{- 1}]$

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(4)

Angular impulse = change in angular momentum.

[Angular impulse] = [Angular momentum] = [ $m v r$ ]

= $[M L^{2} T^{- 1}]$

Q 17 :

If B is magnetic field and $μ_{0}$ is permeability of free space, then the dimensions of ( $B / μ_{0}$ ) is [2025]

$M T^{- 2} A^{- 1}$
$L^{- 1} A$
$L T^{- 2} A^{- 1}$
$M L^{2} T^{- 2} A^{- 1}$

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(2)

Magnetic field at the axis of a long solenoid, $B = μ_{0} n i$

$[\frac{B}{μ_{0}}] = [n i] = L^{— 1} A$

Q 18 :

Which one of the following is the correct dimensional formula for the capacitance in F? M, L, T and C stand for unit of mass, length, time and charge, [2025]

$[F] = [C^{2} M^{- 2} L^{2} T^{2}]$
$[F] = [C M^{- 2} L^{- 2} T^{- 2}]$
$[F] = [C M^{- 1} L^{- 2} T^{2}]$
$[F] = [C^{2} M^{- 1} L^{- 2} T^{2}]$

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(4)

$C = \frac{q}{V} = \frac{q}{V} \cdot \frac{q}{q} = \frac{q^{2}}{Work done}$

$∴ [F] = \frac{[C^{2}]}{[M L^{2} T^{- 2}]} = [C^{2} M^{- 1} L^{- 2} T^{2}]$

Q 19 :

The position of a particle moving on x-axis is given by x(t) = A sin t + B $\cos^{2} t$ + $C t^{2}$ + D, where t is time. The dimension of $\frac{A B C}{D}$ is [2025]

L
$L^{3} T^{- 2}$
$L^{2} T^{- 2}$
$L^{2}$

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(3)

Dimension [x(t)] = [L]

[A] = [L]

[B] = [L]

[C] = [ $L T^{- 2}$ ]

[D] = [L]

$[\frac{A B C}{D}] = [\frac{L \times L \times L T^{- 2}}{L}] = [L^{2} T^{- 2}]$

Q 20 :

The electric flux is $ϕ = α σ + β λ$ , where $λ$ and $σ$ are linear and surface charge density, respectively, $(\frac{α}{β})$ represents [2025]

charge
electric field
displacement
area

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(3)

$ϕ = α σ + β λ$

$[ϕ] = [α σ] = [β λ]$

$[α] = \frac{[ϕ]}{[σ]} [β] = \frac{[ϕ]}{[λ]}$

$\frac{[α]}{[β]} = \frac{[Q / L]}{[Q / Area]} = [\frac{Area}{Length}]$

$[\frac{α}{β}] = L$

Q 21 :

Match List I with List II

List I List II

(A) Permeability of free space I. [ $M L^{2} T^{- 2}$ ]

(B) Magnetic field II. [ $M T^{- 2} A^{- 1}$ ]

(C) Magnetic moment III. [ $M L T^{- 2} A^{- 2}$ ]

(D) Torsional constant IV. [ $L^{2} A$ ]

Choose the correct answer from the options given below : [2025]

(A)-(I), (B)-(IV), (C)-(II), (D)-(III)
(A)-(II), (B)-(I), (C)-(III), (D)-(IV)
(A)-(IV), (B)-(III), (C)-(I), (D)-(II)
(A)-(III), (B)-(II), (C)-(IV), (D)-(I)

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(4)

$B = \frac{μ_{0} I}{2 π r}$

$\Rightarrow [μ_{0}] = [\frac{B \cdot r}{I}] = [\frac{M T^{- 2} A^{- 1} \times L}{A}] = [M L T^{- 2} A^{- 2}]$

magnetic field F = qvB

$B = [\frac{M L T^{- 2}}{A T L / T}] = [M T^{- 2} A^{- 1}]$

$[M] = [L^{2} A]$

$τ = C θ \Rightarrow C = [\frac{τ}{θ}] = [M L^{2} T^{- 2}]$

Q 22 :

The energy E and momentum p of a moving body of mass m are related by some equation. Given that c represents the speed of light, identify the correct equation. [2025]

$E^{2} = p c^{2} + m^{2} c^{4}$
$E^{2} = p c^{2} + m^{2} c^{2}$
$E^{2} = p^{2} c^{2} + m^{2} c^{2}$
$E^{2} = p^{2} c^{2} + m^{2} c^{4}$

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(4)

We need to check the dimensions only.

$[E] = M^{1} L^{2} T^{- 2}$

$[P c] = M^{1} L^{1} T^{- 1} \cdot L^{1} T^{- 1} = M^{1} L^{2} T^{- 2}$

$[m c^{2}] = M^{1} L^{2} T^{- 2}$

So, $E^{2} = p^{2} c^{2} + m^{2} c^{4}$ (dimensionally)

Q 23 :

Match List I with List II

List I List II

A Angular Impulse (I) [ $M^{0} L^{2} T^{- 2}$ ]

B Latent Heat (II) [ $M L^{2} T^{- 3} A^{- 1}$ ]

C Electrical Resistivity (III) [ $M L^{2} T^{- 1}$ ]

D Electromotive Force (IV) [ $M L^{3} T^{- 3} A^{- 2}$ ]

Choose the correct answer from the options given below : [2025]

(A)-(III), (B)-(I), (C)-(IV), (D)-(II)
(A)-(I), (B)-(III), (C)-(IV), (D)-(II)
(A)-(III), (B)-(I), (C)-(II), (D)-(IV)
(A)-(II), (B)-(I), (C)-(IV), (D)-(III)

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(1)

Angular impulse = [ $M L^{2} T^{- 1}$ ]

Latent Heat = [ $M^{0} L^{2} T^{- 2}$ ]

Electrical Resistivity = [ $M L^{3} T^{- 3} A^{- 2}$ ]

Electromotive Force = [ $M L^{2} T^{- 3} A^{- 1}$ ]

Q 24 :

The pair of physical quantities not having same dimensions is [2025]

Torque and energy
Surface tension and impulse
Angular momentum and Planck's constant
Pressure and Young's modulus

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(2)

[Angular momentum] = $M L^{2} T^{- 1}$

[Planck's Constant] = $M L^{2} T^{- 1}$

[Torque] = $M L^{2} T^{- 2}$

[Energy] = $M L^{2} T^{- 2}$

[Surface tension] = $M T^{- 2}$

[Impulse] = $M L T^{- 1}$

[Pressure] = $M L^{- 1} T^{- 2}$

[Young's modulus] = $M L^{- 1} T^{- 2}$

Q 25 :

The expression given below shows the variation of velocity (v) with time (t), $v = A t^{2} + \frac{B t}{C + t}$ . The dimension of ABC is : [2025]

$[M^{0} L^{2} T^{- 3}]$
$[M^{0} L^{1} T^{- 3}]$
$[M^{0} L^{1} T^{- 2}]$
$[M^{0} L^{2} T^{- 2}]$

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(1)

$v = A t^{2} + \frac{B t}{C + t}$

$[v] = [A] t^{2} = [\frac{B t}{C + t}]$

[A] = $L T^{- 3}$

[C] = T

[B] = $L T^{- 1}$

[ABC] = $L^{2} T^{- 3}$

Q 26 :

Match List I with List II.

List I List II

(A) Young's Modulus (I) $M L^{- 1} T^{- 1}$

(B) Torque (II) $M L^{- 1} T^{- 2}$

(C) Coefficient of Viscosity (III) $M^{- 1} L^{3} T^{- 2}$

(D) Gravitational Constant (IV) $M L^{2} T^{- 2}$

Choose the correct answer from the options given below : [2025]

(A)-(I), (B)-(III), (C)-(II), (D)-(IV)
(A)-(II), (B)-(I), (C)-(IV), (D)-(III)
(A)-(IV), (B)-(II), (C)-(III), (D)-(I)
(A)-(II), (B)-(IV), (C)-(I), (D)-(III)

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(4)

(A) $[Y] = \frac{F}{A (\frac{△ l}{l})} \Rightarrow \frac{M L T^{- 2}}{L^{2}} = M L^{- 1} T^{- 2}$

(B) Torque $(\vec{τ}) = \vec{r} \times \vec{F} = L \times M L T^{- 2} = M L^{2} T^{- 2}$

(C) Coefficient of viscosity $\Rightarrow F = η A \frac{d V}{d t}$

$[η] = \frac{M L T^{- 2}}{L^{2}} \times T = M L^{- 1} T^{- 1}$

(D) Gravitational constant (G)

$F = \frac{G M_{1} M_{2}}{r^{2}}$

$[G] = \frac{F \cdot r^{2}}{m_{1} m_{2}} = \frac{M L T^{- 2} \times L^{2}}{M^{2}} = M^{- 1} L^{3} T^{- 2}$

Q 27 :

The equation for real gas is given by $(P + \frac{a}{V^{2}}) (V - b) = R T$ , where P, V, T and R are the pressure, volume, temperature and gas constant, respectively. The dimension of $a b^{- 2}$ is equivalent to that of : [2025]

Planck's constant
Compressibility
Strain
Energy density

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(4)

Given, $[P + \frac{a}{V^{2}}] (V - b) = R T$

$[P] = [\frac{a}{V^{2}}] \Rightarrow [a] = [P] [V^{2}]$

$[a] = [P] [V^{2}] = [M L^{- 1} T^{- 2}] [L^{6}] = M L^{5} T^{- 2}$

$[b] = [V] = L^{3}$

$[a b^{- 2}] = [M L^{5} T^{- 2} L^{- 6}] = [M L^{- 1} T^{- 2}]$

Dimension of energy density.

Q 28 :

Match List I with List II

List I List II

(A) Coefficient of viscosity (I) [ $M L^{0} T^{- 3}$ ]

(B) Intensity of wave (II) [ $M L^{- 2} T^{- 2}$ ]

(C) Pressure gradient (III) [ $M^{- 1} L T^{2}$ ]

(D) Compressibility (IV) [ $M L^{- 1} T^{- 1}$ ]

Choose the correct answer from the options given below : [2025]

(A)-(I), (B)-(IV), (C)-(III), (D)-(II)
(A)-(IV), (B)-(I), (C)-(II), (D)-(III)
(A)-(IV), (B)-(II), (C)-(I), (D)-(III)
(A)-(II), (B)-(III), (C)-(IV), (D)-(I)

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(2)

$η \equiv \frac{F l}{A v} \equiv [M L^{- 1} T^{- 1}]$	A $\to$ IV
$I \equiv \frac{P}{A} \equiv [M T^{- 3}]$	B $\to$ I
$\frac{d P}{d x} \equiv [M L^{- 2} T^{- 2}]$	C $\to$ II
$K \equiv \frac{1}{B} \equiv [M^{- 1} L T^{2}]$	D $\to$ III

Q 29 :

Given a charge q, current I and permeability of vacuum $μ_{0}$ . Which of the following quantity has the dimension of momentum? [2025]

$q I / μ_{0}$
$q μ_{0} I$
$q^{2} μ_{0} I$
$q μ_{0} / I$

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(2)

Momentum = mv

$r = \frac{m v}{q B}$

$[m v] \equiv [q B r] \Rightarrow [m v] \equiv [q \frac{μ_{0} I}{2 π r} r]$

$\Rightarrow [m v] \equiv [q μ_{0} I]$

Q 30 :

If $μ_{0}$ and $ε_{0}$ are the permeability and permittivity of free space, respectively. then the dimension of $(\frac{1}{μ_{0} ε_{0}})$ is : [2025]

$L / T^{2}$
$L^{2} / T^{2}$
$T^{2} / L$
$T^{2} / L^{2}$

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(2)

$C = \frac{1}{\sqrt{μ_{0} ε_{0}}}$

$μ_{0} ε_{0} \equiv \frac{1}{C^{2}} \equiv [L^{- 2} T^{2}]$

$∴ [\frac{1}{μ_{0} ε_{0}}] = \frac{L^{2}}{T^{2}}$

Topic Question Set

Q 1 :

The equation of stationary wave is: [2024]

$y = 2 a \sin (\frac{2 πnt}{λ}) \cos (\frac{2 πx}{λ}) .$

Which of the following is not correct?

Q 2 :

Applying the principle of homogeneity of dimensions, determine which one is correct, where T is time period, G is gravitational constant, M is mass, r is radius of orbit. [2024]

Q 3 :

If $G$ be the gravitational constant and $u$ be the energy density then which of the following quantity have the dimensions as that of the $\sqrt{u G} .$ [2024]

Q 4 :

What is the dimensional formula of $a b^{- 1}$ in the equation $(P + \frac{a}{V^{2}}) (V - b) = R T,$ where letters have their usual meaning. [2024]

Q 6 :

Given below are two statements: [2024]

Statements (I): Dimensions of specific heat is $[L^{2} T^{- 2} K^{- 1}]$

Statements (Il): Dimensions of gas constant is $[M L^{2} T^{- 1} K^{- 1}]$

In the light of the above statements choose the most appropriate answer from the options given below:

Q 7 :

If $\in_{0}$ is the permittivity of free space and E is the electric field, then $\in_{0} E^{2}$ has the dimensions [2024]

Q 8 :

The dimensional formula of latent heat is [2024]

Q 9 :

The de-Broglie wavelength associated with a particle of mass $m$ and energy $E$ is $h / \sqrt{2 m E} .$ The dimensional formula for Planck's constant is [2024]

Q 10 :

Given below are two statements: [2024]

Statement (I): Planck's constant and angular momentum have same dimensions.

Statement (II): Linear momentum and moment of force have same dimensions.

In the light of the above statements, choose the correct answer from the options given below:

Q 11 :

The equation of state of a real gas is given by $(P + \frac{a}{V^{2}}) (V - b) = R T,$ where $P, V$ and $T$ are pressure, volume and temperature respectively and R is the universal gas constant. The dimensions of $\frac{a}{b^{2}}$ is similar to that of: [2024]

Q 12 :

Match List-I with List-II [2024]

List-I List-II

A. Coefficient of viscosity I. $[M L^{2} T^{- 2}]$

B. Surface tension II. $[M L^{2} T^{- 1}]$

C. Angular momentum III. $[M L^{- 1} T^{- 1}]$

D. Rotational kinetic energy IV. $[M L^{0} T^{- 2}]$

Q 13 :

If mass is written as $m = k c^{P} G^{- 1 / 2} h^{1 / 2}$ then the value of $P$ will be (Constants have their usual meaning with $k$ a dimensionless constant) [2024]

Q 14 :

A force is represented by $F = a x^{2} + b t^{1 / 2}$ where $x$ = distance and $t$ = time. The dimensions of $b^{2} / a$ are [2024]

Q 15 :

Consider two physical quantities A and B related to each other as $E = \frac{B - x^{2}}{A t}$ where $E, x$ and $t$ have dimensions of energy, length and time respectively. The dimension of A B is [2024]

Q 16 :

The dimensional formula of angular impulse is [2024]

Q 17 :

If B is magnetic field and $μ_{0}$ is permeability of free space, then the dimensions of ( $B / μ_{0}$ ) is [2025]

Q 18 :

Which one of the following is the correct dimensional formula for the capacitance in F? M, L, T and C stand for unit of mass, length, time and charge, [2025]

Q 19 :

The position of a particle moving on x-axis is given by x(t) = A sin t + B $\cos^{2} t$ + $C t^{2}$ + D, where t is time. The dimension of $\frac{A B C}{D}$ is [2025]

Q 20 :

The electric flux is $ϕ = α σ + β λ$ , where $λ$ and $σ$ are linear and surface charge density, respectively, $(\frac{α}{β})$ represents [2025]

Q 22 :

The energy E and momentum p of a moving body of mass m are related by some equation. Given that c represents the speed of light, identify the correct equation. [2025]

Q 24 :

The pair of physical quantities not having same dimensions is [2025]

Q 25 :

The expression given below shows the variation of velocity (v) with time (t), $v = A t^{2} + \frac{B t}{C + t}$ . The dimension of ABC is : [2025]

Q 27 :

The equation for real gas is given by $(P + \frac{a}{V^{2}}) (V - b) = R T$ , where P, V, T and R are the pressure, volume, temperature and gas constant, respectively. The dimension of $a b^{- 2}$ is equivalent to that of : [2025]

Q 29 :

Given a charge q, current I and permeability of vacuum $μ_{0}$ . Which of the following quantity has the dimension of momentum? [2025]

Q 30 :

If $μ_{0}$ and $ε_{0}$ are the permeability and permittivity of free space, respectively. then the dimension of $(\frac{1}{μ_{0} ε_{0}})$ is : [2025]

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	List I		List II
A	Torque	(I)	$[M^{1} L^{2} T^{- 2} A^{- 2}]$
B	Magnetic Field	(II)	$[L^{2} A^{1}]$
C	Magnetic moment	(III)	$[M^{1} T^{- 2} A^{- 1}]$
D	Permeability of free space	(IV)	$[M^{1} L^{2} T^{- 2}]$

	List-I		List-II
A.	Coefficient of viscosity	I.	$[M L^{2} T^{- 2}]$
B.	Surface tension	II.	$[M L^{2} T^{- 1}]$
C.	Angular momentum	III.	$[M L^{- 1} T^{- 1}]$
D.	Rotational kinetic energy	IV.	$[M L^{0} T^{- 2}]$

	List I		List II
A	Angular Impulse	(I)	[ $M^{0} L^{2} T^{- 2}$ ]
B	Latent Heat	(II)	[ $M L^{2} T^{- 3} A^{- 1}$ ]
C	Electrical Resistivity	(III)	[ $M L^{2} T^{- 1}$ ]
D	Electromotive Force	(IV)	[ $M L^{3} T^{- 3} A^{- 2}$ ]

	List I		List II
(A)	Young's Modulus	(I)	$M L^{- 1} T^{- 1}$
(B)	Torque	(II)	$M L^{- 1} T^{- 2}$
(C)	Coefficient of Viscosity	(III)	$M^{- 1} L^{3} T^{- 2}$
(D)	Gravitational Constant	(IV)	$M L^{2} T^{- 2}$

	List I		List II
(A)	Coefficient of viscosity	(I)	[ $M L^{0} T^{- 3}$ ]
(B)	Intensity of wave	(II)	[ $M L^{- 2} T^{- 2}$ ]
(C)	Pressure gradient	(III)	[ $M^{- 1} L T^{2}$ ]
(D)	Compressibility	(IV)	[ $M L^{- 1} T^{- 1}$ ]

Topic Question Set

Q 1 : The equation of stationary wave is: [2024] y=2asin(2πntλ)cos(2πxλ). Which of the following is not correct?

Q 2 : Applying the principle of homogeneity of dimensions, determine which one is correct, where T is time period, G is gravitational constant, M is mass, r is radius of orbit. [2024]

Q 3 : If G be the gravitational constant and u be the energy density then which of the following quantity have the dimensions as that of the uG. [2024]

Q 4 : What is the dimensional formula of ab-1 in the equation (P+aV2)(V-b)=RT, where letters have their usual meaning. [2024]

Q 5 : Match List-I with List-II: [2024] List I List II A Torque (I) [M1L2T-2A-2] B Magnetic Field (II) [L2A1] C Magnetic moment (III) [M1T-2A-1] D Permeability of free space (IV) [M1L2T-2] Choose the correct answer from the options given below:

Q 6 : Given below are two statements: [2024] Statements (I): Dimensions of specific heat is [L2T-2K-1] Statements (Il): Dimensions of gas constant is [ML2T-1K-1] In the light of the above statements choose the most appropriate answer from the options given below:

Q 7 : If ∈0 is the permittivity of free space and E is the electric field, then ∈0E2 has the dimensions [2024]

Q 8 : The dimensional formula of latent heat is [2024]

Q 9 : The de-Broglie wavelength associated with a particle of mass m and energy E is h/2mE. The dimensional formula for Planck's constant is [2024]

Q 10 : Given below are two statements: [2024] Statement (I): Planck's constant and angular momentum have same dimensions. Statement (II): Linear momentum and moment of force have same dimensions. In the light of the above statements, choose the correct answer from the options given below:

Q 11 : The equation of state of a real gas is given by (P+aV2)(V-b)=RT, where P,V and T are pressure, volume and temperature respectively and R is the universal gas constant. The dimensions of ab2 is similar to that of: [2024]

Q 12 : Match List-I with List-II [2024] List-I List-II A. Coefficient of viscosity I. [ML2T-2] B. Surface tension II. [ML2T-1] C. Angular momentum III. [ML-1T-1] D. Rotational kinetic energy IV. [ML0T-2]

Q 13 : If mass is written as m=kcPG-1/2h1/2 then the value of P will be (Constants have their usual meaning with k a dimensionless constant) [2024]

Q 14 : A force is represented by F=ax2+bt1/2 where x = distance and t = time. The dimensions of b2/a are [2024]

Q 15 : Consider two physical quantities A and B related to each other as E=B-x2At where E,x and t have dimensions of energy, length and time respectively. The dimension of A B is [2024]

Q 16 : The dimensional formula of angular impulse is [2024]

Q 17 : If B is magnetic field and μ0 is permeability of free space, then the dimensions of (B/μ0) is [2025]

Q 18 : Which one of the following is the correct dimensional formula for the capacitance in F? M, L, T and C stand for unit of mass, length, time and charge, [2025]

Q 19 : The position of a particle moving on x-axis is given by x(t) = A sin t + B cos2t + Ct2+ D, where t is time. The dimension of ABCD is [2025]

Q 20 : The electric flux is ϕ=ασ+βλ, where λ and σ are linear and surface charge density, respectively, (αβ) represents [2025]

Q 21 : Match List I with List II List I List II (A) Permeability of free space I. [ML2T–2] (B) Magnetic field II. [MT–2A–1] (C) Magnetic moment III. [MLT–2A–2] (D) Torsional constant IV. [L2A] Choose the correct answer from the options given below : [2025]

Q 22 : The energy E and momentum p of a moving body of mass m are related by some equation. Given that c represents the speed of light, identify the correct equation. [2025]

Q 23 : Match List I with List II List I List II A Angular Impulse (I) [M0L2T–2] B Latent Heat (II) [ML2T–3A–1] C Electrical Resistivity (III) [ML2T–1] D Electromotive Force (IV) [ML3T–3A–2] Choose the correct answer from the options given below : [2025]

Q 24 : The pair of physical quantities not having same dimensions is [2025]

Q 25 : The expression given below shows the variation of velocity (v) with time (t), v=At2+BtC+t. The dimension of ABC is : [2025]

Q 26 : Match List I with List II. List I List II (A) Young's Modulus (I) ML–1T–1 (B) Torque (II) ML–1T–2 (C) Coefficient of Viscosity (III) M–1L3T–2 (D) Gravitational Constant (IV) ML2T–2 Choose the correct answer from the options given below : [2025]

Q 27 : The equation for real gas is given by (P+aV2)(V–b)=RT, where P, V, T and R are the pressure, volume, temperature and gas constant, respectively. The dimension of ab–2 is equivalent to that of : [2025]

Q 28 : Match List I with List II List I List II (A) Coefficient of viscosity (I) [ML0T–3] (B) Intensity of wave (II) [ML–2T–2] (C) Pressure gradient (III) [M–1LT2] (D) Compressibility (IV) [ML–1T–1] Choose the correct answer from the options given below : [2025]

Q 29 : Given a charge q, current I and permeability of vacuum μ0. Which of the following quantity has the dimension of momentum? [2025]

Q 30 : If μ0 and ε0 are the permeability and permittivity of free space, respectively. then the dimension of (1μ0ε0) is : [2025]

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Q 1 :

The equation of stationary wave is: [2024]

$y = 2 a \sin (\frac{2 πnt}{λ}) \cos (\frac{2 πx}{λ}) .$

Which of the following is not correct?

Q 2 :

Applying the principle of homogeneity of dimensions, determine which one is correct, where T is time period, G is gravitational constant, M is mass, r is radius of orbit. [2024]

Q 3 :

If $G$ be the gravitational constant and $u$ be the energy density then which of the following quantity have the dimensions as that of the $\sqrt{u G} .$ [2024]

Q 4 :

What is the dimensional formula of $a b^{- 1}$ in the equation $(P + \frac{a}{V^{2}}) (V - b) = R T,$ where letters have their usual meaning. [2024]

Q 6 :

Given below are two statements: [2024]

Statements (I): Dimensions of specific heat is $[L^{2} T^{- 2} K^{- 1}]$

Statements (Il): Dimensions of gas constant is $[M L^{2} T^{- 1} K^{- 1}]$

In the light of the above statements choose the most appropriate answer from the options given below:

Q 7 :

If $\in_{0}$ is the permittivity of free space and E is the electric field, then $\in_{0} E^{2}$ has the dimensions [2024]

Q 8 :

The dimensional formula of latent heat is [2024]

Q 9 :

The de-Broglie wavelength associated with a particle of mass $m$ and energy $E$ is $h / \sqrt{2 m E} .$ The dimensional formula for Planck's constant is [2024]

Q 10 :

Given below are two statements: [2024]

Statement (I): Planck's constant and angular momentum have same dimensions.

Statement (II): Linear momentum and moment of force have same dimensions.

In the light of the above statements, choose the correct answer from the options given below:

Q 11 :

The equation of state of a real gas is given by $(P + \frac{a}{V^{2}}) (V - b) = R T,$ where $P, V$ and $T$ are pressure, volume and temperature respectively and R is the universal gas constant. The dimensions of $\frac{a}{b^{2}}$ is similar to that of: [2024]

Q 12 :

Match List-I with List-II [2024]

List-I List-II

A. Coefficient of viscosity I. $[M L^{2} T^{- 2}]$

B. Surface tension II. $[M L^{2} T^{- 1}]$

C. Angular momentum III. $[M L^{- 1} T^{- 1}]$

D. Rotational kinetic energy IV. $[M L^{0} T^{- 2}]$

Q 13 :

If mass is written as $m = k c^{P} G^{- 1 / 2} h^{1 / 2}$ then the value of $P$ will be (Constants have their usual meaning with $k$ a dimensionless constant) [2024]

Q 14 :

A force is represented by $F = a x^{2} + b t^{1 / 2}$ where $x$ = distance and $t$ = time. The dimensions of $b^{2} / a$ are [2024]

Q 15 :

Consider two physical quantities A and B related to each other as $E = \frac{B - x^{2}}{A t}$ where $E, x$ and $t$ have dimensions of energy, length and time respectively. The dimension of A B is [2024]

Q 16 :

The dimensional formula of angular impulse is [2024]

Q 17 :

If B is magnetic field and $μ_{0}$ is permeability of free space, then the dimensions of ( $B / μ_{0}$ ) is [2025]

Q 18 :

Which one of the following is the correct dimensional formula for the capacitance in F? M, L, T and C stand for unit of mass, length, time and charge, [2025]

Q 19 :

The position of a particle moving on x-axis is given by x(t) = A sin t + B $\cos^{2} t$ + $C t^{2}$ + D, where t is time. The dimension of $\frac{A B C}{D}$ is [2025]

Q 20 :

The electric flux is $ϕ = α σ + β λ$ , where $λ$ and $σ$ are linear and surface charge density, respectively, $(\frac{α}{β})$ represents [2025]

Q 22 :

The energy E and momentum p of a moving body of mass m are related by some equation. Given that c represents the speed of light, identify the correct equation. [2025]

Q 24 :

The pair of physical quantities not having same dimensions is [2025]

Q 25 :

The expression given below shows the variation of velocity (v) with time (t), $v = A t^{2} + \frac{B t}{C + t}$ . The dimension of ABC is : [2025]

Q 27 :

The equation for real gas is given by $(P + \frac{a}{V^{2}}) (V - b) = R T$ , where P, V, T and R are the pressure, volume, temperature and gas constant, respectively. The dimension of $a b^{- 2}$ is equivalent to that of : [2025]

Q 29 :

Given a charge q, current I and permeability of vacuum $μ_{0}$ . Which of the following quantity has the dimension of momentum? [2025]

Q 30 :

If $μ_{0}$ and $ε_{0}$ are the permeability and permittivity of free space, respectively. then the dimension of $(\frac{1}{μ_{0} ε_{0}})$ is : [2025]