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Bernoulli's principle is given by the equation: \(P+ {1\over 2}ρ v^2 +ρ g h= K (const.)\)

The dimension of P/K is the same as _______.


1. density
2. velocity
3. pressure
4. dimensionless

1 Answer

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Correct Answer - Option 4 : dimensionless

CONCEPT:

  • Bernoulli's Principle: It states the total mechanical energy of the moving fluid comprising the energy associated with the fluid pressure (P), the gravitational potential energy of elevation ρgh, and the kinetic energy of the fluid motion \({1\over 2}ρ v^2 \), remains constant. i.e. 

\(P+ {1\over 2}ρ v^2 +ρ g h= const.\)

Where p is the pressure exerted by the fluid, v is the velocity of the fluid, ρ is the density of the fluid, h is the height of the container.

  • Dimensional formula: It is a compound expression that shows which of the fundamental quantities and how are they involved in making of a physical quantity.
  • The dimension's principle of homogeneity says that an equation is dimensionally correct only if each term has the same dimension on both sides of the equation.

EXPLANATION:

Given equation is \(P+ {1\over 2}ρ v^2 +ρ g h= K (const.)\)

So, the dimension of [P] will be equal to the dimension of [\( {1\over 2}ρ v^2 \)] and [\(ρ g h\)] and \( K (const.)\)

\([P]= [{1\over 2}ρ v^2] =[ρ g h]= [K]\)

The dimension of [P]  = The dimension of [K]

So the dimension of \([{P\over K}]\) = dimensional less or no dimension.

  • There the quantity \({P\over K}\) is a dimensionless quantity.
  • Hence the correct answer is option 4.

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