Dimension of a Physical Quantity

Q 41 :

Match List I with List II:

List I List II

A. Pressure gradient I. [ $M^{0} L^{2} T^{- 2}$ ]

B. Energy density II. [ $M^{1} L^{- 1} T^{- 2}$ ]

C. Electric field III. [ $M^{1} L^{- 2} T^{- 2}$ ]

D. Latent heat IV. [ $M^{1} L^{1} T^{- 3} A^{- 1}$ ]

Choose the correct answer from the options given below: [2023]

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

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(3)

Pressure gradient = $\frac{d p}{d x} = \frac{[M L^{- 1} T^{- 2}]}{[L]} = [M^{1} L^{- 2} T^{- 2}]$

Energy density = $\frac{energy}{volume} = \frac{[M L^{2} T^{- 2}]}{[L^{3}]} = [M^{1} L^{- 1} T^{- 2}]$

Electric field = $\frac{Force}{charge} = \frac{[M L T^{- 2}]}{[A \cdot T]} = [M^{1} L^{1} T^{- 3} A^{- 1}]$

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

Q 42 :

The equation of a circle is given $x^{2} + y^{2} = a^{2}$ , where a is the radius. If the equation is modified to change the origin other than (0, 0), then find out the correct dimensions of A and B in a new equation : ${(x - A t)}^{2} + {(y - \frac{t}{B})}^{2} = a^{2}$

The dimensions of t is given as [ $T^{- 1}$ ]. [2023]

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

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(2)

${(x - A t)}^{2} + {(y - \frac{t}{B})}^{2} = a^{2}$

$[A t] = A \times \frac{1}{T} = L$

$∴ [A] = T^{1} L^{1}$

$\frac{t}{B}$ is in meters

$∴ \frac{1}{T [B]} = L \Rightarrow [B] = T^{1} L^{- 1}$

Q 43 :

Match List I with List II.

List I List II

A. Torque I. $kg m^{- 1} s^{- 2}$

B. Energy density II. $kg {ms}^{- 1}$

C. Pressure gradient III. $kg m^{- 2} s^{- 2}$

D. Impulse IV. $kg m^{2} s^{- 2}$

Choose the correct answer from the options given below: [2023]

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

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(4)

Torque $\to kg m^{2} s^{- 2} (IV)$

Energy density $\to kg m^{- 1} s^{- 2} (I)$

Pressure gradient $\to kg m^{- 2} s^{- 2} (III)$

Impulse $\to kg {ms}^{- 1} (II)$

Q 44 :

Match List I with List II.

List I List II

A. Angular momentum I. [ $M L^{2} T^{- 2}$ ]

B. Torque II. [ $M L^{- 2} T^{- 2}$ ]

C. Stress III. [ $M L^{2} T^{- 1}$ ]

D. Pressure gradient IV. [ $M L^{- 1} T^{- 2}$ ]

Choose the correct answer from the options given below: [2023]

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

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(2)

$\bar{L} = \vec{r} \times \vec{p} \Rightarrow [L] = [M^{0} L^{1} T^{0}] [M^{1} L^{1} T^{- 1}] = [M^{1} L^{2} T^{- 1}]$

$\bar{τ} = \vec{r} \times \vec{F} \Rightarrow [τ] = [L^{1}] [M L T^{- 2}] = [M L^{- 1} T^{- 2}]$

Stress $\equiv$ Pressure = $\frac{F}{A} \Rightarrow [Stress] = [M L^{- 1} T^{- 2}]$

Pressure Gradient = $\frac{d P}{d x}$

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

Q 45 :

$(P + \frac{a}{V^{2}}) (V - b) = R T$ represents the equation of state of some gases. Where P is the pressure, V is the volume, T is the temperature and a, b, R are the constants. The physical quantity, which has dimensional formula as that of $\frac{b^{2}}{a}$ , will be [2023]

bulk modulus
modulus of rigidity
compressibility
energy density

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(3)

[b] = [V]

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

$∴ [\frac{b^{2}}{a}] = \frac{1}{[P]} = \frac{1}{[B]} = [K]$

Q 46 :

If the velocity of light c, universal gravitational constant G and planck's constant h are chosen as fundamental quantities. The dimensions of mass in the new system is: [2023]

$[h^{\frac{1}{2}} c^{- \frac{1}{2}} G^{1}]$
$[h^{1} c^{1} G^{- 1}]$
$[h^{- \frac{1}{2}} c^{\frac{1}{2}} G^{\frac{1}{2}}]$
$[h^{\frac{1}{2}} c^{\frac{1}{2}} G^{- \frac{1}{2}}]$

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(4)

Dimensional formale of mass is $H^{x} C^{y} G^{z}$

$M^{1} = {(M L^{2} T^{- 1})}^{x} (L T^{- 1}) {(M^{- 1} L^{3} T^{- 2})}^{z}$

$M^{1} L^{0} T^{0} = M^{x - z} L^{2 x + y + 3 z} T^{- x - y - 2 z}$

on comparing both side

$x - z = 1$

$2 x + y + 3 z = 0$

$- x - y - 2 z = 0$

On solving above equations we get

$x = \frac{1}{2}, y = \frac{1}{2}, z = \frac{- 1}{2}$

Q 47 :

If R, $X_{L}$ and $X_{C}$ represent resistance, inductive reactance and capacitive reactance. Then which of the following in dimensionless: [2023]

$R X_{L} X_{C}$
$\frac{R}{\sqrt{X_{L} X_{C}}}$
$\frac{R}{X_{L} X_{C}}$
$R \frac{X_{L}}{X_{C}}$

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(2)

All three have same dimension therefore $\frac{R}{\sqrt{X_{L} X_{C}}}$ is dimensionless.

Q 48 :

Match List I with List II

Liast I List II

A. Planck's constant (h) I, [ $M^{1} L^{2} T^{- 2}$ ]

B. Stopping potential ( $V_{S}$ ) II. [ $M^{1} L^{1} T^{- 1}$ ]

C. Work function ( $ϕ$ ) III. [ $M^{1} L^{2} T^{- 1}$ ]

D. Momentum (p) IV. [ $M^{1} L^{2} T^{- 3} A^{- 1}$ ]

Choose the correct answer from the options given below: [2023]

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

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(2)

(A) Planck's constant

hv = E

$h = \frac{E}{v} = \frac{M^{1} L^{2} T^{- 2}}{T^{- 2}} = M^{1} L^{2} T^{- 1}$

(B) E = qv

$V = \frac{E}{q} = \frac{M^{1} L^{2} T^{- 2}}{A^{1} T^{1}} = M^{1} L^{2} T^{- 3} A^{- 1}$

(C) $ϕ$ (work function) = energy = $M^{1} L^{2} T^{- 2}$

(D) Momentum (p) = F.t = $M^{1} L^{1} T^{- 1} T^{1} = M^{1} L^{1} T^{- 1}$

Q 49 :

Dimension of $\frac{1}{μ_{0} ε_{0}}$ should be equal to [2023]

$\frac{L}{T}$
$\frac{L^{2}}{T^{2}}$
$\frac{T}{L}$
$\frac{T^{2}}{L^{2}}$

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(2)

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

Q 50 :

Match List I with List II

List I List II

A. Torque I. $M L^{- 2} T^{- 2}$

B. Stress II. $M L^{2} T^{- 2}$

C, Pressure gradient III. $M L^{- 1} T^{- 1}$

D. Coefficient of viscosity IV. $M L^{- 1} T^{- 2}$

Choose the correct answer from the options given below: [2023]

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

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(2)

A. Torque $\Rightarrow \vec{τ} = \vec{r} \times \vec{F}$

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

B. Stress = $\frac{F}{A} \Rightarrow \frac{M L T^{- 2}}{L^{2}}$

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

C. Pressure gradient = $\frac{△ P}{△ X}$

$\Rightarrow \frac{[F / A]}{[L]} = \frac{M L T^{- 2}}{L^{3}} \Rightarrow M L^{- 2} T^{- 2}$

D. Cofficient of viscosity $\Rightarrow F = 6 π η η r v$

$M L T^{- 2} = [η] L^{2} T^{- 1}$

$[η] = M L^{- 1} T^{- 1}$

Topic Question Set

Q 43 :

Match List I with List II.

List I List II

A. Torque I. $kg m^{- 1} s^{- 2}$

B. Energy density II. $kg {ms}^{- 1}$

C. Pressure gradient III. $kg m^{- 2} s^{- 2}$

D. Impulse IV. $kg m^{2} s^{- 2}$

Choose the correct answer from the options given below: [2023]

Q 44 :

Match List I with List II.

List I List II

A. Angular momentum I. [ $M L^{2} T^{- 2}$ ]

B. Torque II. [ $M L^{- 2} T^{- 2}$ ]

C. Stress III. [ $M L^{2} T^{- 1}$ ]

D. Pressure gradient IV. [ $M L^{- 1} T^{- 2}$ ]

Choose the correct answer from the options given below: [2023]

Q 45 :

$(P + \frac{a}{V^{2}}) (V - b) = R T$ represents the equation of state of some gases. Where P is the pressure, V is the volume, T is the temperature and a, b, R are the constants. The physical quantity, which has dimensional formula as that of $\frac{b^{2}}{a}$ , will be [2023]

Q 46 :

If the velocity of light c, universal gravitational constant G and planck's constant h are chosen as fundamental quantities. The dimensions of mass in the new system is: [2023]

Q 47 :

If R, $X_{L}$ and $X_{C}$ represent resistance, inductive reactance and capacitive reactance. Then which of the following in dimensionless: [2023]

Q 49 :

Dimension of $\frac{1}{μ_{0} ε_{0}}$ should be equal to [2023]

Q 50 :

Match List I with List II

List I List II

A. Torque I. $M L^{- 2} T^{- 2}$

B. Stress II. $M L^{2} T^{- 2}$

C, Pressure gradient III. $M L^{- 1} T^{- 1}$

D. Coefficient of viscosity IV. $M L^{- 1} T^{- 2}$

Choose the correct answer from the options given below: [2023]

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	List I		List II
A.	Pressure gradient	I.	[ $M^{0} L^{2} T^{- 2}$ ]
B.	Energy density	II.	[ $M^{1} L^{- 1} T^{- 2}$ ]
C.	Electric field	III.	[ $M^{1} L^{- 2} T^{- 2}$ ]
D.	Latent heat	IV.	[ $M^{1} L^{1} T^{- 3} A^{- 1}$ ]

	List I		List II
A.	Torque	I.	$kg m^{- 1} s^{- 2}$
B.	Energy density	II.	$kg {ms}^{- 1}$
C.	Pressure gradient	III.	$kg m^{- 2} s^{- 2}$
D.	Impulse	IV.	$kg m^{2} s^{- 2}$

	List I		List II
A.	Angular momentum	I.	[ $M L^{2} T^{- 2}$ ]
B.	Torque	II.	[ $M L^{- 2} T^{- 2}$ ]
C.	Stress	III.	[ $M L^{2} T^{- 1}$ ]
D.	Pressure gradient	IV.	[ $M L^{- 1} T^{- 2}$ ]

	Liast I		List II
A.	Planck's constant (h)	I,	[ $M^{1} L^{2} T^{- 2}$ ]
B.	Stopping potential ( $V_{S}$ )	II.	[ $M^{1} L^{1} T^{- 1}$ ]
C.	Work function ( $ϕ$ )	III.	[ $M^{1} L^{2} T^{- 1}$ ]
D.	Momentum (p)	IV.	[ $M^{1} L^{2} T^{- 3} A^{- 1}$ ]

Topic Question Set

Q 49 : Dimension of 1μ0ε0 should be equal to [2023] .question-row { display: flex; align-items: flex-start; gap: 8px; } .q-no { font-size: 17px; white-space: nowrap; margin-left: -15px; } .q-text { font-size: 18px; flex: 1; } .q-no, .q-text p { line-height: 30px; }

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Q 49 :

Dimension of $\frac{1}{μ_{0} ε_{0}}$ should be equal to [2023]