Question

Which is not a dimensionless quantity is :
A. Moment of momentum
B. Moment of force
C. Moment of inertia
D. All of the above

Answer 1

Hint: A dimensional quantity is a quantity that has no dimensions. That means its dimensional formula is $\left[ {{M}^{0}}{{L}^{0}}{{T}^{0}} \right]$. So check which of the three quantities, given in the options have dimensional formula equal to $\left[ {{M}^{0}}{{L}^{0}}{{T}^{0}} \right]$.

Complete answer:
A dimensionless quantity is a physical quantity that does not have any dimensions.
If a quantity is dimensionless, then its dimensional formula will be equal to $\left[ {{M}^{0}}{{L}^{0}}{{T}^{0}} \right]$.
Let us check which of the options given have dimensions and which of them are dimensionless quantities.

(A) Moment of momentum is a rotational analogy of momentum. It is also called angular momentum. It is equal to the product of moment of inertia (I) and the angular velocity ($\omega $) of a particle about the axis of the rotation.
i.e. $L=I\omega $.
The dimensional formula of moment of momentum is $\left[ M{{L}^{2}}{{T}^{-1}} \right]$.
This means that angular momentum is not a dimensionless quantity.

(B) Moment of force is also called torque. When a force is applied at a point, a torque is generated about an axis. The torque ($\tau $) is defined as the cross product of the force (F) and the position vector (r) of the point from the axis of rotation.
i.e. $\overrightarrow{\tau }=\overrightarrow{r}\times \overrightarrow{F}$ .
The dimensional formula of torque is $\left[ M{{L}^{2}}{{T}^{-2}} \right]$.
This means that torque is not a dimensionless quantity.

(C) Moment of inertia of a particle is defined as the product of the mass (m) of the particle and the square of the perpendicular distance (d) of the particle from the axis of rotation.
i.e. $I=m{{r}^{2}}$
The dimensional formula of moment of inertia is $\left[ M{{L}^{2}}{{T}^{0}} \right]$.
This means that angular momentum is not a dimensionless quantity.
Therefore, no given quantities are dimensionless.

So, the correct answer is “Option D”.

Note:
A good example of a dimensionless quantity is a ratio of two same quantities. Since the ratio is of the same quantities, when we find the dimensional formula of the ratio, the dimensions will cancel each other and there will be no dimension.