Basic Maths and Units and DimensionsMedium
Question
The speed of light (c), gravitational constant (G) and plank′s contant (h) are taken as fundamental units in a system. The dimensio9ns of time in this new system should be
Options
A.G1/2h1/2c-5/2
B.G-1/2h1/2c1/2
C.G1/2h1/2c-3/2
D.G1/2h1/2c1/2
Solution
Let time, T ∝ cxGyhz
⇒ T = kcxGyhz
Taking dimensions on both sides
[MOLOT1] = [LT-1]X[M-1L3T-2]Y[ML2T-1]Z
i.e.,
[MOLOT1] = [M-Y+ZLX+3Y+2ZT-X-2Y-Z]
Equuating power of M,L,T on both sides we get
- y + z = 0 ..........(1)
x + 3y + 2z = 0 ..........(2)
- x - 2y -z = 0 ...........(3)
From (1) ⇒ z = y
Adding (2)and (3) ⇒ y + z = 1
or 2y = 1 (from (1)) i.e.,y =
∴ z = y =
Putting these values in (2) we get
x +
+ 1 = 0 or x = 
Hence, [T] = [G1/2h1/2c-5/2]
⇒ T = kcxGyhz
Taking dimensions on both sides
[MOLOT1] = [LT-1]X[M-1L3T-2]Y[ML2T-1]Z
i.e.,
[MOLOT1] = [M-Y+ZLX+3Y+2ZT-X-2Y-Z]
Equuating power of M,L,T on both sides we get
- y + z = 0 ..........(1)
x + 3y + 2z = 0 ..........(2)
- x - 2y -z = 0 ...........(3)
From (1) ⇒ z = y
Adding (2)and (3) ⇒ y + z = 1
or 2y = 1 (from (1)) i.e.,y =
∴ z = y =
Putting these values in (2) we get
x +
+ 1 = 0 or x = Hence, [T] = [G1/2h1/2c-5/2]
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