Dimension of a Physical Quantity

Q 51 :

If force (F), velocity (V) and time (T) are considered as fundamental physical quantity, then dimensional formula of density will be [2023]

$F V^{- 2} T^{2}$
$F V^{- 4} T^{2}$
$F V^{4} T^{- 6}$
$F^{2} V^{- 2} T^{6}$

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2

3

4

(2)

$[M L^{- 3}] = {[M L T^{- 2}]}^{a} {[L T^{- 1}]}^{b} [T]^{c} = [M^{a} L^{a + b} T^{- 2 a + b + c}]$

a = 1

a + b = –3 $\Rightarrow$ b = –4

Also, – 2a – b + c = 0

c = 2

Q 52 :

Match List-I with List-II

List I List II

A. Spring constant I. ( $T^{- 1}$ )

B. Angular speed II. ( $M T^{- 2}$ )

C. Angular momentum III. ( $M L^{2}$ )

D. Moment of Inertia IV. ( $M L^{2} T^{- 1}$ )

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

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

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(1)

Spring Constant

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

$[ω] = \frac{[θ]}{[t]} = \frac{1}{T} = T^{- 1}$

Q 53 :

In the equation $[X + \frac{a}{Y^{2}}] [Y - b] = R T$ , X is pressure, Y is volume, R is universal gas constant and T is temperature. The physical quantity a equivalent to the ratio $\frac{a}{b}$ is [2023]

Energy
Impulse
Pressure gradient
Coefficient of viscosity

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(1)

X and $\frac{a}{Y^{2}}$ have same dimensions

Y and b have same dimensions

$∴ [a] = [M L^{5} T^{- 2}]$

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

$\frac{[a]}{[b]} = [M L^{2} T^{- 2}]$ has dimensions of energy.

Q 54 :

The speed of a wave produced in water is given by $v = λ^{a} g^{b} ρ^{c}$ . When $λ$ , g and $ρ$ are wavelength of wave, acceleration due to gravity and density of water respectively. The values of a, b and c respectively, are [2023]

$\frac{1}{2}, \frac{1}{2}, 0$
1, 1, 0
1, –1, 0
$\frac{1}{2}, 0, \frac{1}{2}$

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(1)

$v = λ^{a} g^{b} ρ^{c}$

using dimension formula

$\Rightarrow [M^{0} L^{1} T^{- 1}] = {[L^{1}]}^{a} {[L^{1} T^{- 2}]}^{b} {[M^{1} L^{- 3}]}^{c}$

$\Rightarrow [M^{0} L^{1} T^{- 1}] = [M^{c} L^{a + b - 3 c} T^{- 2 b}]$

$∴ c = 0, a + b - 3 c = 1, - 2 b = - 1 \Rightarrow b = \frac{1}{2}$

Now, a + b –3c = 1

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

$\Rightarrow a = \frac{1}{2}$

$∴ a = \frac{1}{2}, b = \frac{1}{2}, c = 0$

Topic Question Set

Q 51 :

If force (F), velocity (V) and time (T) are considered as fundamental physical quantity, then dimensional formula of density will be [2023]

Q 52 :

Match List-I with List-II

List I List II

A. Spring constant I. ( $T^{- 1}$ )

B. Angular speed II. ( $M T^{- 2}$ )

C. Angular momentum III. ( $M L^{2}$ )

D. Moment of Inertia IV. ( $M L^{2} T^{- 1}$ )

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

Q 53 :

In the equation $[X + \frac{a}{Y^{2}}] [Y - b] = R T$ , X is pressure, Y is volume, R is universal gas constant and T is temperature. The physical quantity a equivalent to the ratio $\frac{a}{b}$ is [2023]

Q 54 :

The speed of a wave produced in water is given by $v = λ^{a} g^{b} ρ^{c}$ . When $λ$ , g and $ρ$ are wavelength of wave, acceleration due to gravity and density of water respectively. The values of a, b and c respectively, are [2023]

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	List I		List II
A.	Spring constant	I.	( $T^{- 1}$ )
B.	Angular speed	II.	( $M T^{- 2}$ )
C.	Angular momentum	III.	( $M L^{2}$ )
D.	Moment of Inertia	IV.	( $M L^{2} T^{- 1}$ )