Question

Assuming that the mass m of the largest stone that can be moved by a flowing river depends upon the velocity v of the water, its density ρ, and the acceleration due to gravity g. Then m is directly proportional to​

Answer 1

Answer:

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Answer 2

Answer:

      m ∝ v⁶.ρ¹.g⁻³

Explanation:

m ∝ vᵃ.ρᵇ.gˣ

vᵃ = [L¹T⁻¹]ᵃ

ρᵇ = [M¹L⁻³]ᵇ

gˣ = [L¹T⁻²]ˣ

Using dimensional analysis

[M¹L⁰T⁰] ∝ [L¹T⁻¹]ᵃ.[M¹L⁻³]ᵇ. [L¹T⁻²]ˣ

[M¹L⁰T⁰] ∝ [Lᵃ⁻³ᵇ⁺ˣ].[T⁻ᵃ⁻²ˣ].[M]ᵇ

[M¹] ∝ [M]ᵇ

b = 1

[L⁰] ∝[Lᵃ⁻³ᵇ⁺ˣ]

a - 3b + x = 0

a - 3 + x = 0

a + x = 3 -----{i}

[T⁰] ∝ [T⁻ᵃ⁻²ˣ]

-a - 2x = 0 ----{ii}

add {i} and {ii}

a + x - a - 2x = 3

x - 2x = 3

-x = 3

x = -3

a - 3 = 3

a = 6

b = 1 , x = -3 , a = 6

m ∝ vᵃ.ρᵇ.gˣ

m ∝ v⁶.ρ¹.g⁻³