Basic Maths and Units and DimensionsHard
Question
Planck′s constant h, speed of light c and gravitational constant G are used to form a unit of length L and a unit of mass M. Then the correct option(s) is(are) :-
Options
A.M ∝ √c
B.M ∝ √G
C.L ∝ √h
D.L ∝ √G
Solution
[h] = [ML2T-1]
[c] = [LT-1]
[G] = [M-1L3T-2]
For unit of length
Let L ∝ hx cy Gz
⇒ From principle of homogeneity
[LHS] = [RHS]
⇒ [M0LT0] = [ML2T-1]x [LT-1]y [M-1L3T-2]z
M0LT0 = Mx-z L2x + y + 3z T - x - y - 2z
comparing we get
x - z = 0
2x + y + 3z = 1
- x - y - 2z = 0
Solving we get, x =
L =
For unit of mass
Let M ∝ hx cy Gz
Solving in similar manner as above
We get x =
M =
[c] = [LT-1]
[G] = [M-1L3T-2]
For unit of length
Let L ∝ hx cy Gz
⇒ From principle of homogeneity
[LHS] = [RHS]
⇒ [M0LT0] = [ML2T-1]x [LT-1]y [M-1L3T-2]z
M0LT0 = Mx-z L2x + y + 3z T - x - y - 2z
comparing we get
x - z = 0
2x + y + 3z = 1
- x - y - 2z = 0
Solving we get, x =
L =
For unit of mass
Let M ∝ hx cy Gz
Solving in similar manner as above
We get x =
M =
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