Bernoulli’s Theorem Statement and its Derivation
In this post, we will discuss in detail the Explanation of Bernoulli’s principle, its statement, and Proof of its Formula through Derivation. Burnoulli’s theorem is an important topic of the class 11 fluid chapter. Burnoulli’s theorem always occurs in half or yearly examinations. So let’s get started with its statement:-
Burnoulli’s Equation and its Diagram
According to Bernoulli, the sum of kinetic energy, potential energy, and pressure energy at unit volume remains constant at every point of the flow is called Burnoulli’s Equation.
P+1/2ρv²+ρgh = Constant
Proof or Derivation of Bernoulli’s Theorem
Work = Force x Displacement
Work at per unit time
Work = Force x Displacement/time
Work = Force x Velocity – 1
Pressure = Force/Area
Force = Pressure x Area
Put in Equation 1
Work = Pressure x Area x Velocity
Change in work = Wf – Wi
= P₁ A₁ V₁ – P₂ A₂ V₂
ΔW = P₁ A₁ V₁ – P₂ A₂ V₂
According to Eqn of Continuity-
A₁ V₁ = m/ρ
ΔW = P₁ m/ρ – P₂ m/ρ
ΔW = m/ρ ( P₁ – P₂)
Change in Potential Energy-
ΔP = Pf – Pi
ΔP = mgh₁ – mgh₂
ΔP = mg (h₁ – h₂)
Change in Kinetic Energy-
ΔK = Kf – Ki
ΔK = 1/2mv₁² – 1/2mv₂ ²
ΔK = 1/2m(v₁² – v₂ ²)
Change in Work = Change in Kinetic and Potential Energy
m/ρ ( P₁ – P₂) = mg (h₁ – h₂) + 1/2m(v₁² – v₂ ²)
Cut all the Masses
1/ρ ( P₁ – P₂) = 1/g (h₁ – h₂) + 1/2(v₁² – v₂ ²)
Rho will move to RHS
P₁ – P₂ = ρgh₁ – ρgh₂ + (1/2ρv₁² – 1/2ρv₂ ²)
P + ρgh + 1/2ρv² = Constant
Hence Proved
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