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

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

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

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

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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(B) $[V] = [b]$ 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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(C) $[\vec{τ}] = [\vec{r} \times \vec{F}] = [M L^{2} T^{- 2}]$

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

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

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

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

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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(C) 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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(B) 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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(C) $Q = m L . \Rightarrow [L] = \frac{[Q]}{[m]}$

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

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

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

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

$[τ] = M L^{2} T^{- 2}$

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

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:

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

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:

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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: