Which one of the following pairs of quantities has no length in their dimension?
[amp_mcq option1=”Surface tension and angular momentum” option2=”Surface tension and strain” option3=”Angular momentum and mass density” option4=”Pressure gradient and angle” correct=”option2″]
This question was previously asked in
UPSC CISF-AC-EXE – 2019
– **Surface tension:** Defined as force per unit length. Dimension is [Force]/[Length] = [MLT⁻²]/[L] = [MT⁻²]. It has no length dimension.
– **Angular momentum:** Defined as the product of moment of inertia and angular velocity (Iω) or cross product of position and linear momentum (r x p). Dimension is [ML²] * [T⁻¹] = [ML²T⁻¹]. It has a length dimension.
– **Strain:** Defined as the ratio of change in dimension to the original dimension (e.g., change in length / original length). Dimension is [L]/[L] = [Dimensionless]. It has no length dimension.
– **Mass density:** Defined as mass per unit volume. Dimension is [M]/[L³] = [ML⁻³]. It has a length dimension.
– **Pressure gradient:** Defined as change in pressure per unit distance. Dimension is [Pressure]/[Length]. Pressure is [Force]/[Area] = [MLT⁻²]/[L²] = [ML⁻²T⁻²]. So, pressure gradient is [ML⁻²T⁻²]/[L] = [ML⁻³T⁻²]. It has a length dimension.
– **Angle:** Defined as the ratio of arc length to radius. Dimension is [L]/[L] = [Dimensionless]. It has no length dimension.
Now let’s check the pairs:
A) Surface tension ([MT⁻²]) and angular momentum ([ML²T⁻¹]). Angular momentum has length.
B) Surface tension ([MT⁻²]) and strain ([Dimensionless]). Neither has length.
C) Angular momentum ([ML²T⁻¹]) and mass density ([ML⁻³]). Both have length.
D) Pressure gradient ([ML⁻³T⁻²]) and angle ([Dimensionless]). Pressure gradient has length.
The pair with no length dimension is Surface tension and Strain.