Question

Select the pair whose dimensions are the same

• A. Pressure and stress
• B. Stress and strain
• C. Pressure and force
• D. Power and force

Answer 1

The correct option is A

Pressure and stress

The explanation of the correct option:

In case of option A

Pressure is defined as physical force applied perpendicular to the unit area of the object.
So, we can write it in the following mathematics form: $Pressure=\frac{Appliedforce}{Area}$

But force applied on an object is nothing but the production of a mass of the object and the applied acceleration on the object.
Dimension of Mass is denoted by $\left[M\right]$
But acceleration is a change in velocity divided by the time applied to the object to move along.

$$\begin{gathered} a=\frac{velocity}{time} \\ a=\frac{{\displaystyle \frac{d}{t}}}{t}\left[As,velocity=\frac{dis\tan ce}{time}\right] \\ a=\frac{d}{t^2} \end{gathered}$$

Dimension of distance is denoted by $\left[L\right]$ and that of time is denoted by $\left[T\right]$.
So, the dimension of acceleration would become =$\frac{L}{T^2}=\left[L{T}^{-2}\right]$

$$\begin{gathered} Dimensionofforce=dimensionofmass\times dimensionofacceleration \\ DIiesnionofforce=\left[ML{T}^{-2}\right] \end{gathered}$$

The 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 =$\left[L^2\right]$

Dimension of pressure =$\frac{dimensionofforce}{dimensionofarea}=\frac{\left[ML{T}^{-2}\right]}{\left[L^2\right]}=\left[M{L}^{-1}{T}^{-2}\right]$

The restoring force applied per unit area of a material is known as stress.

We can derive the following equation for stress: $Stress=\frac{Forceapplied}{Area}$

Dimension of the stress =$\frac{dimensionofforce}{dimensionofarea}=\frac{\left[ML{T}^{-2}\right]}{\left[L^2\right]}=\left[M{L}^{-1}{T}^{-2}\right]$

Hence, dimension of pressure = dimension of stress

The explanation of the incorrect option:

In case of option B

Dimension of the stress =$\frac{dimensionofforce}{dimensionofarea}=\frac{\left[ML{T}^{-2}\right]}{\left[L^2\right]}=\left[M{L}^{-1}{T}^{-2}\right]$

Strain is defined as the ratio of change in dimension divided by the original dimension of the material.

So, the dimension of strain = $\frac{changeindimension}{initialdimension}=\frac{\left[L\right]}{\left[L\right]}=\left[M^0{L}^0{T}^0\right]$

In case of option C

Dimension of pressure=Dimension of stress =$\left[M{L}^{-1}{T}^{-2}\right]$

Dimension of force$=\left[ML{T}^{-2}\right]$

In case of option D

Power is defined as the work done on a physical body per unit of time.
Work done is measured by the production of force applied to a body and the displacement happens due to the force applied. $workdone=(Force\times displacement)=(F\times S)$

So, $Power=\frac{work}{time}=\frac{FS}{t}$

We have already calculated the dimension of force, so we will just put these values to find the dimension of power.

So the dimension of powerr=$\frac{dimensionofwork}{dimensionoftimee}=\frac{\left[ML{T}^{-2}\right]\times \left[L\right]}{\left[T\right]}=\left[M{L}^2{T}^{-3}\right]$

Dimension of force$=\left[ML{T}^{-2}\right]$

Hence, Option A is correct.