Question

If surface tension (S), Moment of inertia (l) and Planck's constant (h), were to be taken as the fundamental units, the dimensional formula for linear momentum would be:

A)\[{{\mathbf{S}}^{{\mathbf{3}}/{\mathbf{2}}}}{{\mathbf{I}}^{{\mathbf{1}}/{\mathbf{2}}}}{{\mathbf{h}}^{\mathbf{0}}}\]

B)\[{{\mathbf{S}}^{{\mathbf{1}}/{\mathbf{2}}}}{{\mathbf{I}}^{{\mathbf{1}}/{\mathbf{2}}}}{{\mathbf{h}}^{\mathbf{0}}}\]

C)\[{{\mathbf{S}}^{{\mathbf{1}}/{\mathbf{2}}}}{{\mathbf{I}}^{{\mathbf{1}}/{\mathbf{2}}}}{{\mathbf{h}}^{ - {\mathbf{1}}}}\] \[\;\;\]

D)\[{{\mathbf{S}}^{{\mathbf{1}}/{\mathbf{2}}}}{{\mathbf{I}}^{{\mathbf{3}}/{\mathbf{2}}}}{{\mathbf{h}}^{ - {\mathbf{1}}}}\]

Answer 1

Hint: For the measurement of a physical quantity we have to evolve a standard for the measurement so that different measurements of same physical quantity can be expressed relative to each other. That standard is called a unit of that physical quantity.
Dimensional formula of a physical quantity is the formula which tells us how and which of the fundamental units has been used for the measurement of that quantity.

Complete step by step solution:
\[Dimensional{\text{ }}formula{\text{ }}for{\text{ }}Surface{\text{ }}tension{\text{ }}\left( s \right){\text{ }} = {\text{ }}M{T^{ - 2}}\]
\[Dimensional{\text{ }}formula{\text{ }}for{\text{ }}Planck's{\text{ }}constant\left( h \right){\text{ }} = {\text{ }}M{L^2}{T^{ - 1}}\]
Also, surface tension (S), Moment of inertia (l) and Planck's constant (h), were to be taken as the fundamental units
Therefore,
\[
  p = {{\mathbf{S}}^a}{{\mathbf{I}}^b}{{\mathbf{h}}^c} \\
  ML{T^{ - 1}} = {\left( {M{T^{ - 2}}} \right)^a}\left( {M{L^2}} \right){\left( {M{L^2}{T^{ - 1}}} \right)^C} \\
  \begin{array}{*{20}{l}}
  {a + b + c{\text{ }} = 12} \\
  {b + 2c{\text{ }} = 1} \\
  { - 2a - c{\text{ }} = - 1} \\
  {a = 21\;b = 21\;c = 0}
\end{array} \\
 \]
So, the dimensional formula for linear momentum if surface tension (S), Moment of inertia (l) and Planck's constant (h), were to be taken as the fundamental units is \[{{\mathbf{S}}^{{\mathbf{1}}/{\mathbf{2}}}}{{\mathbf{I}}^{{\mathbf{1}}/{\mathbf{2}}}}{{\mathbf{h}}^{\mathbf{0}}}\].

Hence, the correct option is B.

Note: Dimensions of a physical quantity are the powers to which the fundamental units are raised to obtain one unit of that quantity.
The fundamental units are the units of the fundamental quantities, and are defined by the International System of Units. They are not dependent upon any other units, and all other units are derived from them.
There are basically seven fundamental units:
1. Length - meter (m)
2. Time - second (s)
3. Amount of substance - mole (mole)
4. Electric current - ampere (A)
5. Temperature - Kelvin (K)
6. Luminous intensity - candela (cd)
7. Mass - kilogram (kg)