Question 39: Units , Dimensions and Error Analysis

Question

$$ 	ext { Match the LIST-I with LIST-II } $$

List-I
List-II

A.
Spring constant
I.

M

L

2

T

−
2

K

−
1

M

L

2

T

−
2

K

−
1

ML^(2)T^(-2)K^(-1)

B.
Thermal conductivity
II.

ML

0

T

−
2

ML

0

T

−
2

ML^(0)T^(-2)

C.
Boltzmann constant
III.

M

L

2

T

−
3

A

−
2

M

L

2

T

−
3

A

−
2

ML^(2)T^(-3)A^(-2)

D.
Inductive reactance
IV.

M

L

T

−
3

K

−
1

M

L

T

−
3

K

−
1

MLT^(-3)K^(-1)

Choose the correct answer from the options given below:

Accepted Answer

A. Spring constant (k)
According to Hooke's Law, the restoring force F of a spring is directly proportional to its displacement x :
$$ \mathrm{F}=-\mathrm{kx} $$
Taking the magnitude to find dimensions :
$$ \mathrm{k}=\frac{\mathrm{F}}{\mathrm{x}} \Rightarrow[\mathrm{k}]=\left[\frac{\mathrm{F}}{\mathrm{x}}ight]=\frac{\left[\mathrm{MLT}^{-2}ight]}{\left[\mathrm{M}^0 \mathrm{LT}^0ight]} $$
$$ [\mathrm{k}]=\frac{\left[\mathrm{MLT}^{-2}ight]}{[\mathrm{L}]}=\left[\mathrm{ML}^{1-1} \mathrm{~T}^{-2}ight]=\left[\mathrm{ML}^0 \mathrm{~T}^{-2}ight] $$
A Ã¢ÂÂ II.
B. Thermal conductivity (K)
From Law of heat conduction, the rate of heat flow $\left(\mathrm{H}=\frac{\mathrm{d} \mathrm{Q}}{\mathrm{dt}}ight)$ is given by:
$$ H=\frac{K A \Delta T}{L} $$
Where A is area, $\Delta \mathrm{T}$ is temperature difference, and L is length.
$$ \mathrm{K}=\frac{\mathrm{H} \cdot \mathrm{~L}}{\mathrm{~A} \cdot \Delta \mathrm{~T}} $$
The dimensions of rate of heat flow, $[\mathrm{H}]=\frac{[	ext { Energy }]}{[	ext { Time }]}=\frac{[	ext { Force ÃÂ distance }]}{[	ext { Time }]}=\frac{\left[\mathrm{MLT}^{-2} 	imes \mathrm{L}ight]}{[\mathrm{T}]}=\left[\mathrm{ML}^2 \mathrm{~T}^{-3}ight]$
So the dimensions of thermal conductivity is,
$[\mathrm{K}]=\frac{\left[\mathrm{ML}^2 \mathrm{~T}^{-3}ight] \cdot[\mathrm{L}]}{\left[\mathrm{L}^2ight] \cdot[\mathrm{K}]}=\frac{\left[\mathrm{ML}^3 \mathrm{~T}^{-3}ight]}{\left[\mathrm{L}^2 \mathrm{~K}ight]}$
$$\Rightarrow $$ $$ [\mathrm{K}]=\left[\mathrm{ML}^{3-2} \mathrm{~T}^{-3} \mathrm{~K}^{-1}ight]=\left[\mathrm{MLT}^{-3} \mathrm{~K}^{-1}ight] $$
B Ã¢ÂÂ IV.
C. Boltzmann constant ( $\mathrm{k}_{\mathrm{B}}$ )
The average kinetic energy (E) of a gas molecule is related to its absolute temperature (T) by :
$$ \mathrm{E}=\frac{3}{2} \mathrm{k}_{\mathrm{B}} \mathrm{~T} $$
The dimensionless constant is $\frac{3}{2}$.
$$ \left[\mathrm{k}_{\mathrm{B}}ight]=\left[\frac{\mathrm{E}}{\mathrm{~T}}ight] $$
We know Energy has the same dimensions as Work ( $\mathrm{F} 	imes \mathrm{d}$ ):
$$ [\mathrm{E}]=\left[\mathrm{ML}^2 \mathrm{~T}^{-2}ight] $$
So, the dimensions of Boltzmann constant is,
$$ \left[\mathrm{k}_{\mathrm{B}}ight]=\frac{\left[\mathrm{ML}^2 \mathrm{~T}^{-2}ight]}{[\mathrm{K}]}=\left[\mathrm{ML}^2 \mathrm{~T}^{-2} \mathrm{~K}^{-1}ight] $$
$$ \mathrm{C} ightarrow \mathrm{I} . $$
D. Inductive reactance ( $\mathrm{X}_{\mathrm{L}}$ )
Inductive reactance is the opposition to alternating current by an inductor. It acts exactly like resistance, so it must have the same dimensional formula as electrical resistance (R).
From Ohm's Law: $\mathrm{V}=\mathrm{IR}$, so $\mathrm{R}=\frac{\mathrm{V}}{\mathrm{I}}$.
Potential is Work done per unit charge (W/q):
$$ [\mathrm{V}]=\frac{[	ext { Work }]}{[	ext { Charge }]}=\frac{\left[\mathrm{ML}^2 \mathrm{~T}^{-2}ight]}{[\mathrm{A} \cdot \mathrm{~T}]}=\left[\mathrm{ML}^2 \mathrm{~T}^{-3} \mathrm{~A}^{-1}ight] $$
So, the dimensions of inductive reactance, i.e., resistance is,
$$ \left[\mathrm{X}_{\mathrm{L}}ight]=[\mathrm{R}]=\frac{[\mathrm{V}]}{[\mathrm{I}]}=\frac{\left[\mathrm{ML}^2 \mathrm{~T}^{-3} \mathrm{~A}^{-1}ight]}{[\mathrm{A}]} $$
$$ \begin{array}{ll}  & {\left[\mathrm{X}_{\mathrm{L}}ight]=\left[\mathrm{ML}^2 \mathrm{~T}^{-3} \mathrm{~A}^{-2}ight]} \ \mathrm{D} ightarrow \mathrm{III} . & \end{array} $$
Therefore, the correct match is, $\mathrm{A} ightarrow$ II, $\mathrm{B} ightarrow$ IV, $\mathrm{C} ightarrow \mathrm{I}, \mathrm{D} ightarrow$ III.
Hence, the correct option is (D).

Answer

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

Answer

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

Answer

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

Answer

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

Units , Dimensions and Error Analysis

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