According to newton’s Law of gravitation, any two particles of the universe attract each other with a force which is proportional to the product of the masses of the particles and inversely proportional to the square of distance between them. If and are masses of two particles separated by distance r then force between them is given as , where G is called universal gravitational constant.According to Max Planck the light is collection of photons such that energy of each photon is where is frequency of the photon and h is called Planck’s constant.According to Einstein if m is mass of a particle (or sometime called dynamic mass of the particle) then its energy is given by the relation where c is speed of light.If [G], [h] and [c] are taken as fundamental quantities then dimensions of mass is:
Question
According to newton’s Law of gravitation, any two particles of the universe attract each other with a force which is proportional to the product of the masses of the particles and inversely proportional to the square of distance between them. If and are masses of two particles separated by distance r then force between them is given as , where G is called universal gravitational constant.According to Max Planck the light is collection of photons such that energy of each photon is where is frequency of the photon and h is called Planck’s constant.According to Einstein if m is mass of a particle (or sometime called dynamic mass of the particle) then its energy is given by the relation where c is speed of light.If [G], [h] and [c] are taken as fundamental quantities then dimensions of mass is:
Solution
The given fundamental quantities are [G], [h] and [c].
- [G] represents the universal gravitational constant. Its dimensions are [M^-1 L^3 T^-2].
- [h] represents Planck's constant. Its dimensions are [M L^2 T^-1].
- [c] represents the speed of light. Its dimensions are [L T^-1].
We are asked to find the dimensions of mass [M] in terms of these fundamental quantities.
From the given relations, we can write the dimensions of mass as:
[M] = [G]^-1 [h] [c]^-2
Substituting the dimensions of [G], [h] and [c] in the above equation, we get:
[M] = ([M^-1 L^3 T^-2])^-1 [M L^2 T^-1] ([L T^-1])^-2 = [M^1 L^-3 T^2] [M L^2 T^-1] [L^-2 T^2] = [M^2 L^-3 T^2] [M L^2 T^-1] = [M^2]
So, the dimensions of mass in terms of the given fundamental quantities are [M^2].
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