Question

What are the dimensions of mass in the dimensional formula of acceleration?
$\begin{array}{c}\text{A}\text{. 1}\\ \text{B}\text{. 2}\\ \text{C}\text{. }{\displaystyle \frac{1}{2}}\\ \text{D}\text{. 0}\end{array}$

Answer 1

Hint: The acceleration can be connected to the fundamental quantities through Newton's second law of motion. If we know the dimensional formula of force and mass then we can obtain the dimensions of acceleration as well and check the dimensions of mass in it.

Complete answer:
Newton's second law of motion gives the following relation between force, mass and acceleration.
$\begin{array}{rl}& F=ma\\ & a={\displaystyle \frac{F}{m}}\end{array}$
We know that the dimensional formula for force is given by the following expression: $\left[ML{T}^{-2}\right]$ while for mass, we have simply $\left[M\right]$ or $\left[M{L}^0{T}^0\right]$. Inserting these values in the above relation, we get the dimensions of acceleration to be
${\displaystyle \frac{\left[ML{T}^{-2}\right]}{\left[M\right]}}=\left[M^{0}L{T}^{-2}\right]$
Therefore, we can say that the dimension of mass in the dimensional formula of acceleration is zero.

So, the correct answer is “Option D.

Additional Information:
Dimensional formula: A dimensional formula of a physical quantity is an expression which describes the dependence of that quantity on the fundamental quantities.
All physical quantities can be expressed in terms of certain fundamental quantities. Following table contains the fundamental quantities and their units and dimensional notation respectively.

No.	Quantities	unit	Dimensional formula
1.	Length	metre (m)	$\left[M^0{L}^1{T}^0\right]$
2.	Mass	kilogram (g)	$\left[M^1{L}^0{T}^0\right]$
3.	Time	second (s)	$\left[M^0{L}^0{T}^1\right]$
4.	Electric current	ampere (A)	$\left[M^0{L}^0{T}^0{A}^1\right]$
5.	Temperature	kelvin [K]	$\left[M^0{L}^0{T}^0{K}^1\right]$
6.	Amount of substance	mole [mol]	$\left[M^0{L}^0{T}^{0}mo{l}^1\right]$
7.	Luminous intensity	candela [cd]	$\left[M^0{L}^0{T}^{0}C{d}^1\right]$

Note:
1. Mass, length and time are most commonly encountered fundamental quantities so they must be specified in all dimensional formulas. The square bracket notation is used only for dimensional formulas.
2. We can obtain the dimensions of acceleration also by using the relation that acceleration is equal to velocity divided by time.