Question

Match list I with II and select the correct answer:

List I	List II
Spring constant	${M^1}{L^2}{T^{ - 2}}$
Pascal	${M^0}{L^0}{T^{ - 1}}$
Hertz	${M^1}{L^0}{T^{ - 2}}$
Joule	${M^1}{L^{ - 1}}{T^{ - 2}}$

(A) ${\text{A - 3, B - 4, C - 2, D - 1}}$
(B) ${\text{A - 4, B - 3, C - 1, D - 2}}$
(C) ${\text{A - 4, B - 3, C - 2, D - 1}}$
(D) ${\text{A - 3, B - 4, C - 1, D - 2}}$

Answer 1

Hint: To select the correct option we must know how to convert the unit of physical quantity to dimensional formula. Find out the physical quantity to which the given unit belongs, and then convert the units to a simple base SI unit and apply the dimensional formula.

Complete step by step solution:
Dimensional Analysis:
Dimensions of a physical quantity are the powers raised to fundamental quantities.
If we follow the following steps correctly we can write the dimensional formula of a physical quantity.
Firstly, we have to check whether the given physical quantity is in SI unit system, if not we have to change the given unit system to SI unit system. If the unit of given physical quantity is not in simple base units, change it to fundamental quantity units. Write the equivalent dimensional formula to the base units. The dimensional formula cannot be used to an equation having more than three physical quantities.

Fundamental quantity	Dimension
Length	L
Mass	M
Time	T
Temperature	K
Electric current	A
Luminous intensity	cd
Amount of substance	mol

The above table shows the fundamental quantities and their dimensions. Using these seven quantities we can convert a given unit to dimension.
Given, Spring constant: the spring constant is derived from Hooke's law. The Hooke's law says that the force required to stretch an elastic object (in case of a spring) is directly proportional to the extension of the spring.
$ \Rightarrow F = - kx$
The spring constant is a proportional constant. It is denoted by k. it is the measure of the spring's stiffness. I.e. when a spring is stretched or compressed its length changes by a quantity from its equilibrium length. The measure of the ability to change its elasticity is known as its stiffness. $ \Rightarrow k = - \dfrac{F}{x}$
Force is the product of mass and acceleration
$ \Rightarrow k = - \dfrac{{m \times a}}{x}{\text{ }}\left[ {\dfrac{{kg \times m{s^{ - 2}}}}{m} = kg{\text{ }}{s^{ - 2}}} \right]$
F is the force
K is the spring constant
X is the change in length
Since spring constant is given by force per change in length, the SI unit of spring constant is $N{m^{ - 1}}$. Which can also be written as $Kg{\text{ }}{{\text{s}}^{ - 2}}$. According to the table the equivalent dimensional formula of $Kg{\text{ }}{{\text{s}}^{ - 2}}$ is ${M^1}{T^{ - 2}}{\text{ or }}{M^1}{L^0}{T^{ - 2}}$.
Pascal: Pascal is the SI unit of pressure used as the unit of internal pressure, stress, Young's modulus, ultimate tensile strength. It is named after French scientist Blaise Pascal, since pressure is defined as force by area.
$ \Rightarrow P = \dfrac{F}{A}$
P is pressure
F is force
A is area
Force is the product of mass and acceleration

$ \Rightarrow P = \dfrac{{m \times a}}{A}{\text{ }}\left[ {\dfrac{{kg \times m{s^{ - 2}}}}{{{m^2}}} = kg{\text{ }}{{\text{m}}^{ - 1}}{\text{ }}{s^{ - 2}}} \right]$
Pascal can be written as $N{m^{ - 1}}$ and also \[{\text{ }}kg{\text{ }}{{\text{m}}^{ - 1}}{\text{ }}{s^{ - 2}}\]. The equivalent dimensional formula of \[{\text{ }}kg{\text{ }}{{\text{m}}^{ - 1}}{\text{ }}{s^{ - 2}}\] according to the table is ${M^1}{L^{ - 1}}{S^{ - 2}}$
Hertz: hertz is the SI unit of frequency. Frequency is defined as the number of rotations or cycles per second. One hertz is defined as one cycle per second. So the hertz can also be written as per second $[{s^{ - 1}}]$. The equivalent dimensional formula of \[{s^{ - 1}}\] according to the table is ${S^{ - 2}}{\text{ or }}{M^0}{L^0}{S^{ - 2}}$
Joule: Joule is the SI unit of energy which is the measure of the capacity to do work or generate heat. The joule is also used as a unit of thermal energy. Sometimes few electric terms are also written in joule. We work is defined as the product of force and displacement.
Since,$W = F \times d$, force is the product of mass and acceleration.
$ \Rightarrow W = m \times a \times d{\text{ }}\left[ {kg \times {\text{m}}{{\text{s}}^{ - 2}} \times m} \right]$
So, Joule can be written as newton per meter, $N{m^{ - 1}}$ or kilogram square meter per square second $kg{\text{ }}{{\text{m}}^2}{\text{ }}{{\text{s}}^{ - 2}}$. The equivalent dimensional formula of \[{\text{ }}kg{\text{ }}{{\text{m}}^2}{\text{ }}{s^{ - 2}}\] according to the table is ${M^1}{L^2}{S^{ - 2}}$
From the above discussion we can match the dimension with their physical quantities

Hence the correct answer is option (A) ${\text{A - 3, B - 4, C - 2, D - 1}}$

Note: Some quantities do not possess dimensions which are called dimensionless quantities. Examples are strain, angle. Etc. They are dimensionless because they are ratios of two quantities having the same dimensional formula. Also the equation with exponential and trigonometric functions cannot be converted to dimensional formula