Question

Select the pair whose dimensions are same:
a. Pressure and stress
b. Stress and Strain
c. Pressure and Force
d. Power and Force

Answer 1

Hint: Here we have to find out the dimension of the given pairs in the option. So we will find out one by one and then we will conclude which pair is the same. Some definitions are used to find out the dimensions and particular terms.

Complete step by step answer:
Pressure is defined as physical force applied perpendicular on the unit area of the object.
So, we can write it in following mathematics form:
\[{\text{Pressure = }}\dfrac{{{\text{Applied force}}}}{{{\text{Area}}}}\] .
But force applied on an object is nothing but the production of mass of the object and the applied acceleration on the object.
Dimension of Mass is denoted by \[[M]\].
But acceleration is change in velocity divided by the time applied on the object to move along.
\[{\text{Acceleration = }}\dfrac{{{\text{velocity}}}}{{{\text{time}}}}\] .

But\[{\text{Velocity = }}\dfrac{{{\text{Distance}}}}{{{\text{time}}}}\]
So, \[{\text{Acceleration = }}\dfrac{{\dfrac{{{\text{distance}}}}{{{\text{time}}}}}}{{{\text{time}}}}{\text{ = }}\dfrac{{{\text{distance}}}}{{{\text{tim}}{{\text{e}}^{\text{2}}}}}\] .
Dimension of distance is denoted by \[[L]\] and that of time is denoted by \[[T]\] .
So, the dimension of acceleration would become \[ = \dfrac{L}{{{T^2}}} = [L{T^{ - 2}}]\] .
Dimension of force will be \[ = \](dimension of mass \[ \times \] dimension of acceleration)\[ = [M \times L{T^{ - 2}}] = [ML{T^{ - 2}}]\]
Area of any shape is defined by its length or radius of the shape.
So, the dimension of the area is nothing but the square of the length.
Dimension of area \[ = [{L^2}]\] .

Dimension of pressure\[{\text{ = }}\dfrac{{{\text{dimension of force}}}}{{{\text{dimension of area}}}}{\text{ = }}\dfrac{{{\text{[ML}}{{\text{T}}^{{\text{ - 2}}}}{\text{]}}}}{{{\text{[}}{{\text{L}}^{\text{2}}}{\text{]}}}}{\text{ = [M}}{{\text{L}}^{{\text{ - 1}}}}{{\text{T}}^{{\text{ - 2}}}}{\text{]}}\] .
The restoring force applied per unit area of a material is known as stress.

We can derive the following equation for stress:
Stress \[{\text{ = }}\dfrac{{{\text{Force applied}}}}{{{\text{Area}}}}\]
Dimension of the stress will be\[{\text{ = }}\dfrac{{{\text{dimension of force}}}}{{{\text{dimension of area}}}}\] .
Dimension of stress\[ = \dfrac{{[ML{T^{ - 2}}]}}{{[{L^2}]}} = [M{L^{ - 1}}{T^{ - 2}}]\] .
Strain is defined as the ratio of change in dimension divided by the original dimension of the material.

So, dimension of strain will be\[{\text{ = }}\dfrac{{{\text{change in dimension}}}}{{{\text{initial dimension}}}}{\text{ = }}\dfrac{{{\text{[L]}}}}{{{\text{[L]}}}}\] .
So, strain has dimension or it is dimensionless quantity.
Power is defined as the work done on a physical body per unit time.
Work done is measured by the production of force applied on a body and the displacement happens due to the force applied.
So, work done \[ = (Force \times displacement) = (F \times S)\] .
So, Power \[ = \dfrac{{work}}{{time}} = \dfrac{{FS}}{t}\] .
So, dimension of the power \[{\text{ = }}\dfrac{{{\text{dimension of work}}}}{{{\text{dimension of time}}}}\] .

We have already calculated the dimension of force, so we will just put these values to find the dimension of power.
So, dimension of power \[ = \dfrac{{[ML{T^{ - 2}}] \times [L]}}{{[T]}} = [M{L^2}{T^{ - 3}}]\] .
So, by comparing the dimensions of pressure, power, stress, strain and force, we can say that only the dimension of pressure and stress are the same.
Dimension of pressure\[ = \] Dimension of stress \[ = \]\[[M{L^{ - 1}}{T^{ - 2}}]\].

Hence, the correct answer is option (A).

Note: Dimension of any quantity is defined by its powers to which the fundamental units are raised to obtain one unit of that quantity.
Every measurement has two parts - one is numbers and another one is unit.
So, proper measurement is requiring defining the dimension actual.