Question

The dimensional formula of physical quantity is $\left[ {{M^a}{L^b}{T^c}} \right]$. 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

Hint: The dimensional formula is an equation which gives relation between the fundamental units and derived units in terms of dimensions. The basic three units are length which is represented by L, mass represented by M and time taken as T.

Formula used:
The formula of the surface tension is given by,
$ \Rightarrow \sigma = \dfrac{F}{L}$
Where force is F and length is L.
The formula of the force is equal to,
$ \Rightarrow F = ma$
Where mass is m and acceleration is a.
The formula of the angular frequency is given by,
$ \Rightarrow \omega = \dfrac{{2\pi }}{T}$
Where the time period is T.
The formula of the spring force is given by,
$F = - k \times x$
Where force is F, the spring constant is k and change of length x.

Complete step by step solution:
It asked in the problem about the dimensional formula of the physical quantity and the terms a, b and c. We will take each physical quantity and find its dimensional formula of each and then find the correct answer for this problem.
Let's consider the physical quantity surface tension.
$ \Rightarrow \sigma = \dfrac{F}{L}$
Where surface tension is$\sigma $, force is F and the length is L.
Dimensional formula of the surface tension is equal to,
$ \Rightarrow \sigma = \dfrac{F}{L}$
$ \Rightarrow \sigma = F \times {L^{ - 1}}$
Force is equal to $kgm{s^{ - 2}}$and length is m.
$ \Rightarrow \sigma = F \times {L^{ - 1}}$
$ \Rightarrow \sigma = kgm{s^{ - 2}} \times {m^{ - 1}}$
$ \Rightarrow \sigma = kg{s^{ - 2}}$
Kg is unit of mass M and second is unit of time T.
$ \Rightarrow \sigma = kg{s^{ - 2}}$
$ \Rightarrow \sigma = \left[ {M{L^0}{T^{ - 2}}} \right]$
Here a=1, b=0 and c=-2. As the option does not match with option A and therefore is not the correct answer.
Now let us consider the option B which is force.
The formula of the force is given by,
$ \Rightarrow F = ma$
Where mass is represented in kg and acceleration is represented in$m{s^{ - 2}}$.
$ \Rightarrow F = ma$
$ \Rightarrow F = kg \times \dfrac{m}{{{s^2}}}$
The mass is kg which is M, the length is in meters i.e. L and time is in seconds i.e. T.
$ \Rightarrow F = kg \times \dfrac{m}{{{s^2}}}$
$ \Rightarrow F = M \times \dfrac{L}{{{T^2}}}$
$ \Rightarrow F = \left[ {ML{T^{ - 2}}} \right]$.
Here a=1, b=1 and c=-2. The values a, b and c do not match with the values in the option B and therefore it is not the correct option.
Let us consider the physical quantity angular frequency. The formula of the angular frequency is given by,
$ \Rightarrow \omega = \dfrac{{2\pi }}{T}$
Where angular frequency is $\omega $ and the time taken is T.
The dimension of the time period is T and as the term $2\pi $ does not have the dimension and therefore we can drop it.
$ \Rightarrow \omega = \dfrac{1}{T}$
$ \Rightarrow \omega = \left[ {{T^{ - 1}}} \right]$
$ \Rightarrow \omega = \left[ {{M^0}{L^0}{T^{ - 1}}} \right]$
Here a=0, b=0 and c=-1. The values of the a, b and c are the same as the given values in the problem and therefore option C is the correct option.
Let us consider the physical quantity spring constant.
The formula of the spring force is given by,
$ \Rightarrow F = - k \times x$
Where force is F, the spring constant is k and the change in the length is x.
$ \Rightarrow F = - k \times x$
$ \Rightarrow k = - \dfrac{F}{x}$
The dimension of force is $kgm{s^{ - 2}}$ the dimension of length is meters (m).
$ \Rightarrow k = - \dfrac{{kgm{s^{ - 2}}}}{m}$
The kg is unit of mass, meter (m) is the unit of length and second is unit of time.
$ \Rightarrow k = - \dfrac{{kgm{s^{ - 2}}}}{m}$
$ \Rightarrow k = - \dfrac{{ML{T^{ - 2}}}}{L}$
We can remove the negative sign as it does not affect the dimensional formula.
$ \Rightarrow k = - \dfrac{{ML{T^{ - 2}}}}{L}$
$ \Rightarrow k = \left[ {{M^1}{L^0}{T^{ - 2}}} \right]$
Here a=0, b=0 and c=-2.
The values of a, b and c are not the same with the values given in the problem and therefore is not the correct answer.

The correct answer for this problem is option C, angular frequency as the values of a, b and c.

Note: The students are advised to understand and remember the formula of surface tension, force, angular frequency and spring force as it is very useful in solving such problems also force is represented in newtons and it has dimension $kgm{s^{ - 2}}$.