Dimension of a Physical Quantity

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

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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(C) (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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(A) $m = K C^{P} G^{- 1 / 2} h^{1 / 2}$ ...(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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(B) $F = a x^{2} + b t^{1 / 2}$

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

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

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

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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(B) $[B] = L^{2}$

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

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

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

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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(D) Angular impulse = change in angular momentum.

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

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

Topic Question Set

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]

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

Topic Question Set

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]

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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]