find the dimensions of linear momentum and surface tension in term of velocity v , density p and frequency v as fundamental quantity
Answers
For Linear Momentum,
Dimensions of the Velocity[v] = Dimension of the Length/Dimension of the Time.
= L/T
= LT⁻¹
Dimension of the Frequency[f] = 1/Dimension of the Time
= 1/T
= T⁻¹
Dimension of the Density[d] = Dimension of Mass/Dimension of the Volume
= M/L³
= ML⁻³
Dimension of Linear Momentum[p] = Dimension of (Mass × Velocity]
= MLT⁻¹
Now, Let the Relation between the Momentum, Velocity, Density and Frequency be
p = vᵃ dᵇ fⁿ
Putting the Dimension of the Quantities.
MLT⁻¹ = [LT⁻¹]ᵃ [ML⁻³]ᵇ [T⁻¹]ⁿ
MLT⁻¹ = LᵃT⁻ᵃ Mᵇ L⁻³ᵇ T⁻ⁿ
MLT⁻¹ = Mᵇ Lᵃ⁻³ᵇ T⁻ⁿ⁻ᵃ
On comparing,
b = 1,
a - 3b = 1
⇒ a - 3(1) = 1
⇒ a = 3 +1 = 4
Also, -n - a = -1
-n -4 = -1
n = 1 - 4
n = -3
Thus, The Relation will be,
p = v⁴ d/f³
For the Surface Tension,
Let the Relation will be ⇒
γ = vᵃ dᵇ fⁿ
Dimension of the Surface tension is MLT⁻²
MLT⁻² = [LT⁻¹]ᵃ [ML⁻³]ᵇ [T⁻¹]ⁿ
∴ MLT⁻² = LᵃT⁻ᵃ Mᵇ L⁻³ᵇ T⁻ⁿ
MLT⁻² = Mᵇ Lᵃ⁻³ᵇ T⁻ⁿ⁻ᵃ
On comapring,
b = 1
a - 3b = 1
a - 3 = 1 ⇒ a = 4
-n -a = -2
-n - 4 = -2
n = -2
Hence, the Relation is γ = v⁴d/f²
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Answer:
Explanation:
For Linear Momentum,
Dimensions of the Velocity[v] = Dimension of the Length/Dimension of the Time.
= L/T
= LT⁻¹
Dimension of the Frequency[f] = 1/Dimension of the Time
= 1/T
= T⁻¹
Dimension of the Density[d] = Dimension of Mass/Dimension of the Volume
= M/L³
= ML⁻³
Dimension of Linear Momentum[p] = Dimension of (Mass × Velocity]
= MLT⁻¹
Now, Let the Relation between the Momentum, Velocity, Density and Frequency be
p = vᵃ dᵇ fⁿ
Putting the Dimension of the Quantities.
MLT⁻¹ = [LT⁻¹]ᵃ [ML⁻³]ᵇ [T⁻¹]ⁿ
MLT⁻¹ = LᵃT⁻ᵃ Mᵇ L⁻³ᵇ T⁻ⁿ
MLT⁻¹ = Mᵇ Lᵃ⁻³ᵇ T⁻ⁿ⁻ᵃ
On comparing,
b = 1,
a - 3b = 1
⇒ a - 3(1) = 1
⇒ a = 3 +1 = 4
Also, -n - a = -1
-n -4 = -1
n = 1 - 4
n = -3
Thus, The Relation will be,
p = v⁴ d/f³
For the Surface Tension,
Let the Relation will be ⇒
γ = vᵃ dᵇ fⁿ
Dimension of the Surface tension is MLT⁻²
MLT⁻² = [LT⁻¹]ᵃ [ML⁻³]ᵇ [T⁻¹]ⁿ
∴ MLT⁻² = LᵃT⁻ᵃ Mᵇ L⁻³ᵇ T⁻ⁿ
MLT⁻² = Mᵇ Lᵃ⁻³ᵇ T⁻ⁿ⁻ᵃ
On comapring,
b = 1
a - 3b = 1
a - 3 = 1 ⇒ a = 4
-n -a = -2
-n - 4 = -2
n = -2
Hence, the Relation is γ = v⁴d/f²