Question

Match list I with II and select the correct answer

List I	List II
a. Spring Constant	1.${M^1}{L^2}{T^{ - 2}}$
b. Pascal	2.${M^0}{L^0}{T^{ - 1}}$
c. Hertz	3.${M^1}{L^0}{T^{ - 2}}$
d .Joule	4.${M^1}{L^{ - 1}}{T^{ - 2}}$

(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: Spring constant can be defined as force that causes a displacement of one unit to the spring.
Pascal is the unit of pressure which is the force acting per unit area.
Hertz is the unit of frequency. It is the number of waves that pass a fixed place in a given amount of time.
Joule is the unit of energy which defines the capacity for doing work.

Complete answer:
According to Hooke’s law
$F = - kx$
Where F represents restoring force
k represents spring constant
X is the displacement of spring
$
  k = \dfrac{F}{x} \\
   \Rightarrow k = \dfrac{{ma}}{x} \\
   \Rightarrow k = \dfrac{{\left[ M \right] \times \left[ {L{T^{ - 2}}} \right]}}{{\left[ L \right]}} \\
   \Rightarrow k = \left[ {M{T^{ - 2}}} \right] \\
 $
We know that pressure is force per unit area.
$
  P = \dfrac{F}{A} \\
   \Rightarrow P = \dfrac{{\left[ {ML{T^{ - 2}}} \right]}}{{\left[ {{L^2}} \right]}} \\
   \Rightarrow P = \left[ {M{L^{ - 1}}{T^{ - 2}}} \right] \\
 $
Frequency can be dimensionally represented as
$
  f = \dfrac{1}{T} \\
   \Rightarrow f = \dfrac{1}{{\left[ {{M^0}{L^0}{T^1}} \right]}} \\
   \Rightarrow f = \left[ {{M^0}{L^0}{T^{ - 1}}} \right] \\
 $
Energy is the ability to do work. The dimensional formula for energy can be calculated form Einstein’s formula
$E = m{c^2}$
$ \Rightarrow E = \left[ {M{{\left( {L{T^{ - 1}}} \right)}^2}} \right]$
$ \Rightarrow E = M{L^{ - 2}}{T^{ - 2}}$
Now we go to the dimensional formulas for all the equations.

The correct answer is option C.

Additional Information:
The quantities that are dependent on other quantities are called fundamental quantities and which are derived from other quantities are called derived quantities. Units which are used to measure fundamental quantities are called fundamental units and which are used to measure derived quantities are called derived units.
There are seven fundamental quantities which are namely length, mass, time, electric current, thermodynamic temperature, intensity of light, quantity of substance.

Note:
Dimensions are the fundamental units raised to obtain one unit of that quantity.
Dimensional formula shows the powers to which powers to which fundamental units are raised to obtain one unit of derived physical quantity is called dimensional formula.
Dimensionless quantities are quantities without units like angle.