If the velocity of light C, coefficient of viscosity(η) gravitational constant(G) are taken as fundamental quantities then the expression of mass(m) in terms of dimensions of these quantities is proportional to
a) c^5/ηG^2
b) η^3G/C^2
c) G^2/ηC^3
d) η^2/C^3G
Answers
Given info : If the velocity of light C, coefficient of viscosity(η) gravitational constant(G) are taken as fundamental quantities..
To find : the expression of mass in term of dimensions of these quantities is proportional to....
Solution : we know, dimension of velocity, C = [LT¯¹]
dimension of viscosity, η = [ML¯¹T¯¹]
dimension of gravitational constant, G = [M¯¹L³T¯²]
dimension of mass, m = [M]
Now let m is proportional to C^x η^y G^z
⇒m = k C^x η^y G^z [ k is proportionality constant]
⇒[M] = k[LT¯¹]^x [ML¯¹T¯¹]^y [M¯¹L³T¯²]^z
⇒[M] = k[M^(y - z) L^(x -y + 3z) T^(-x -y - 2z)]
on comparing we get,
y - z = 1 .....(1)
x - y + 3z = 0 ⇒x + 3z = y ......(2)
-x - y - 2z = 0 ⇒x + y +2z = 0 .....(3)
From equations (2) and (3) we get,
2y = z now putting it in equation (1) we get, y = -1 , z = -2 and x = 5
So, m = kC⁵/ηG²
Therefore the correct option is option (a) C⁵/ηG²