Question

If light velocity (c), gravitational acceleration (g) and atmospheric pressure (p) are taken as basic units then the length parameter ....

Answer 1

It is given that, Velocity of light (c) , gravitational acceleration (g) and Atmospheric pressure (p) are taken as basic units.

So, We have to find the dimension of Length.

Let the dimension formula of Length parameter in terms of Velocity of light (c) , gravitational acceleration (g) and atmospheric pressure (p) be cˣ gʸ pᶻ .

So, According to the homogeneity principle,

⇒ [L] = cˣ gʸ pᶻ

⇒ [L] = [LT⁻¹]ˣ [LT⁻²]ʸ [ML⁻¹T⁻²]ᶻ

⇒ [L] = [M]ᶻ [L]ˣ⁺ʸ⁻ᶻ [T]⁻ˣ⁻²ʸ⁻²ᶻ

⇒ [M⁰L¹T⁰] = [M]ᶻ [L]ˣ⁺ʸ⁻ᶻ [T]⁻ˣ⁻²ʸ⁻²ᶻ

On Comparing both sides,

1. [M⁰] = [Mᶻ]

∴ z = 0 ...(1)

2. [L¹] = [L]ˣ⁺ʸ⁻ᶻ

⇒ x + y - z = 1

⇒ x + y = 1 [ from (1) ] ...(2)

3. [T⁰] = [T]⁻ˣ⁻²ʸ⁻²ᶻ

⇒ -x - 2y - 2z = 0

⇒ -x - 2y = 0 [ from (1) ]

⇒ x + 2y = 0 ...(3)

Subtracting (2) from (3),

⇒ x + 2y - (x + y) = 0 - 1

⇒ x + 2y - x - y = -1

⇒ y = -1 ...(4)

Substituting value of y in (2), we get

⇒ x + (-1) = 1

⇒ x = 2 ...(5)

Now,

⇒ [L] = cˣ gʸ pᶻ

From (1), (4) and (5),

⇒ [L] = [c²g⁻¹p⁰]

Answer 2

Answer is given below :)

It is given that Velocity of light c , gravitational acceleration g and atmospheric pressure p are taken as basic units.

So, we have to find the dimension of the length.

Let, dimension formula of length parameter in the term Rock of Velocity of light c , gravitational acceleration g and atmospheric pressure p be c^x , g^y , p^z

So, according to the homogeneity principle,

⚫L = c^x g^y p^z

⚫L = [LT-¹]^x [LT-²]^y [ML-¹T-²]^z

⚫L = [M]^z [L]^x+y-z [T]-x-²y -²z

⚫M⁰L¹T⁰ = [M]^z [L]^x+y-z [T]-x-²y-²z

On comparing both sides,

_________________________

[M⁰]=[M²]

Therefore, z = 0

_________________________

[L¹] = [L]^x+y-z

x+y-z = 1

x+y = 1

____________________________

T⁰ = [T]^-x -2y-2z

-x - 2y - 2z = 0

-x -2y = 0

x + 2y = 0

____________________________

Subtract 2 from 3

x + 2y - x + y = 0 - 1

x + 2y - x - y = -1

y = -1

____________________________

Substituting value of y in (2) we get

x + (-1) = 1

x = 2

____________________________

Now,

[L] = c^x , g^y , p^z

From 1 , 4 and 5

[L] = c² g-¹ p⁰

Hope it's helpful

Thank you :)