Answer

Verified

67.8k+ views

**Hint**We solve this problem by using the method of dimensional analysis. We find the dimensional formula of the qualities mentioned in the question. We equate the product of these dimensional formulas to the dimensional formula of mass. Solving the equation and finding the powers we get the dimensional formula of mass in terms of speed of light, force, and kinetic energy.

**Complete step by step answer:**

Speed of light has units of \[m/s\](meters per second)

Its dimensional formula is

\[C = [L{T^{ - 1}}]\]

The units of force are newton or \[kgm/{s^2}\] (kg meter per second square)

The dimensional formula of force is

\[F = [{M^1}{L^1}{T^{ - 2}}]\]

Unit of kinetic energy is $kg{m^2}/{s^2}$(kg meter square per second square or joules)

The dimensional formula of kinetic energy is

$K = [{M^1}{L^2}{T^{ - 2}}]$

Here,

Mass, Length, time are represented by$M,L,T$ respectively

The mass has a dimensional formula of

$M = [{M^1}]$

Equating the dimensional formula of mass with the product of other quantities

\[M = {C^\alpha }{F^\beta }{K^\gamma }\].....(1)

\[[{M^1}] = {[L{T^{ - 1}}]^\alpha }{[{M^1}{L^1}{T^{ - 2}}]^\beta }{[{M^1}{L^2}{T^{ - 2}}]^\gamma }\]

\[\Rightarrow [{M^1}] = [{M^{\beta + \gamma }}{L^{\alpha + \beta + 2\gamma }}{T^{ - \alpha - 2\beta - 2\gamma }}]\]

Solving for the powers

\[ \;\beta + \gamma = 1\]

\[\alpha + \beta + 2\gamma = 0\]

\[\alpha + 2\beta + 2\gamma = 0\]

Solving the above three equations,

$\beta = 0$

$\gamma = 1$

$\alpha = - 2$

Substituting the powers back in equation (1)

\[[M] = [{C^{ - 2}}{K^1}]\]

**Hence option (A) $K{C^{ - 2}}$is the correct answer.**

**Additional information**An equation, which gives the relation between fundamental units and derived units in terms of dimensions is called dimensional formula. In terms of mechanics length, mass, time are taken as the fundamental units.

**Note**Dimensional analysis can be very useful to solve any problem. Using dimensional analysis, we can find the units of any quantity. Dimensional analysis can be very handy even for cross-checking the final answer units. This way we can be sure that the answer we found is correct.

Recently Updated Pages

Write an article on the need and importance of sports class 10 english JEE_Main

Choose the exact meaning of the given idiomphrase The class 9 english JEE_Main

Choose the one which best expresses the meaning of class 9 english JEE_Main

What does a hydrometer consist of A A cylindrical stem class 9 physics JEE_Main

A motorcyclist of mass m is to negotiate a curve of class 9 physics JEE_Main

A man stands at a distance of 250m from a wall He shoots class 9 physics JEE_Main

Other Pages

Electric field due to uniformly charged sphere class 12 physics JEE_Main

Differentiate between homogeneous and heterogeneous class 12 chemistry JEE_Main

A vector of 10N makes an angle of 30circ with positive class 11 physics JEE_Main

If a wire of resistance R is stretched to double of class 12 physics JEE_Main