Question

The dimensional formula of physical quantity is ${{M}^{a}}{{L}^{b}}{{T}^{c}}$. Then that physical quantity is
A. Surface tension if a=1, b=1, c=-2
B. Force if a=1, b=1, c=2
C. Angular frequency if a=0, b=0, c=-1
D. Spring constant if a=1, b=-1, c=-2

Answer 1

Define definition of quantities given in options from which formula can be obtained. Put the value of all the quantities present in the formula by considering the value of every quantity as 1 unit. Arrange the formula in standard formula given as $Q=\left[ {{M}^{a}}{{L}^{b}}{{T}^{c}} \right]$ . Compare the value of a, b, c and get the correct physical quantity.

Complete answer:
For convenience, the base quantities are represented by one letter symbols. Generally, mass is denoted by M, length by L, time by T. The physical quantity that is expressed in terms of the base quantities is enclosed in square brackets to remind that the equation is among the dimensions and not among the magnitudes. Such an expression, for a physical quantity in terms of the base quantities is called the dimensional formula.
The dimensional formula of a quantity is given by
$Q=\left[ {{M}^{a}}{{L}^{b}}{{T}^{c}} \right]$
Consider option A:
Surface tension is defined as force per unit length acting at right angle to an imaginary line drawn on the free surface of liquid.
Formula of surface tension is given by
$T=\dfrac{F}{l}$
S.I. unit is N/m.
Therefore its dimension is $\left[ {{M}^{1}}{{L}^{0}}{{T}^{-2}} \right]$
Equate it with $\left[ {{M}^{a}}{{L}^{b}}{{T}^{c}} \right]$ i.e.
$\left[ {{M}^{a}}{{L}^{b}}{{T}^{c}} \right]=\left[ {{M}^{1}}{{L}^{0}}{{T}^{-2}} \right]$
$a=1,b=0,c=-2$
Since the value of $a,b,c$ is not matching.
Therefore option A is incorrect.
Consider option B:
Force is defined a product of mass and acceleration
Formula of force is given by
$F=ma$
S.I. unit is N.
Therefore its dimension is $\left[ {{M}^{1}}{{L}^{1}}{{T}^{-2}} \right]$
Equate it with $\left[ {{M}^{a}}{{L}^{b}}{{T}^{c}} \right]$ i.e.
$\left[ {{M}^{a}}{{L}^{b}}{{T}^{c}} \right]=\left[ {{M}^{1}}{{L}^{1}}{{T}^{-2}} \right]$
$a=1,b=1,c=-2$
Since the value of a,b,c is not matching.
Therefore option B is incorrect.
Now consider option C:
Angular frequency is defined as reciprocal of time.
Formula of angular frequency is given by
$f=\dfrac{1}{T}$
S.I. unit is ${{\sec }^{-1}}$ .
Therefore its dimension is $\left[ {{M}^{0}}{{L}^{0}}{{T}^{-1}} \right]$
Equate it with $\left[ {{M}^{a}}{{L}^{b}}{{T}^{c}} \right]$ i.e.
$\left[ {{M}^{a}}{{L}^{b}}{{T}^{c}} \right]=\left[ {{M}^{0}}{{L}^{0}}{{T}^{-1}} \right]$ , since value of a,b,c is matching i.e. $a=0,b=0,c=-1$ . Therefore option C is correct.
Consider option D:
Spring constant is defined as the force required to stretch or compress a spring by a distance that spring gets elongated or shorter.
Formula of spring constant is given by,
$k=\dfrac{F}{x}$
S.I. unit is N/m.
Therefore its dimension is $\left[ {{M}^{1}}{{L}^{0}}{{T}^{-2}} \right]$
Equate it with $\left[ {{M}^{a}}{{L}^{b}}{{T}^{c}} \right]$ i.e.
$\left[ {{M}^{a}}{{L}^{b}}{{T}^{c}} \right]=\left[ {{M}^{1}}{{L}^{0}}{{T}^{-2}} \right]$
$a=1,b=0,c=-2$
Since the value of a,b,c is not matching.
Therefore option D is incorrect.

Therefore option C is the correct option.

Note:
To calculate the dimension of quantity, you don’t need to memorize it. You just need to understand the definition and the formula of definition from which you can define or guess the formula. By using formula, defined units of the formula and arrange it in standardize form $Q=\left[ {{M}^{a}}{{L}^{b}}{{T}^{c}} \right]$. In this question, we have calculated dimension using the formula $f=\dfrac{1}{T}$ . You can also define the dimension of angular frequency using another formula like $\omega =2\pi \nu$. Hence we can say that the physical quantity is angular frequency as angular frequency has dimensions of ${{T}^{-1}}$.