Question

The Bernoulli’s equation is given by
\[P + \dfrac{1}{2}\rho {v^2} + \rho gh = k\]
Where, \[P\]=pressure, \[\rho \]=density, \[v\]=speed, \[h\]=height of the liquid column, \[g\]=acceleration due to gravity and \[k\] is constant. The dimensional formula for \[k\] is same as that for:
A. Velocity gradient
B. Pressure gradient
C. Modulus of elasticity
D. Thrust

Answer 1

Hint: Recall the principle of homogeneity of dimensions for an equation. Also, recall the condition for addition of dimensions of two physical quantities. Determine the dimensions of constant \[k\] and check which of the given physical quantities in the options have the dimensions same as that of \[k\].

Complete answer:
The Bernoulli’s equation is given by
\[P + \dfrac{1}{2}\rho {v^2} + \rho gh = k\]

Here, we have asked to determine the dimensional formula for \[k\] and check which of the physical quantities given in the options has same dimensional formula as \[k\].

We know that for the sum of the dimensions of any two physical quantities, the dimensions of those two physical quantities must be the same. Also, according to the homogeneity principle of dimensions, the dimensions on the right hand side and left hand side of an equation should have the same dimensions.

Hence, each term on the left hand side and the term on the right hand side of Bernoulli's equation should have the same dimensions. So, we can say that the dimensions of \[k\] are the same as that of the pressure \[P\]. The dimensions of pressure are \[\left[ {M{L^{ - 1}}{T^{ - 2}}} \right]\].

The velocity gradient is the small change in velocity for small change in displacement. The dimensions of velocity gradient are \[\left[ {{T^{ - 1}}} \right]\] which are not equal to the dimensions of \[k\]. Hence, the option A is incorrect.

The pressure gradient is the small change in pressure for small change in distance. The dimensions of velocity gradient are \[\left[ {M{L^{ - 2}}{T^{ - 2}}} \right]\] which are not equal to the dimensions of \[k\]. Hence, option B is incorrect.

The thrust is the upward force acting on an object floating on the liquid. The dimensions of thrust are \[\left[ {ML{T^{ - 2}}} \right]\] which are not equal to the dimensions of \[k\]. Hence, the option D is incorrect.

The modulus of elasticity of a material is the ratio of longitudinal stress and longitudinal strain acting on the material. The dimensions of modulus of elasticity are \[\left[ {M{L^{ - 1}}{T^{ - 2}}} \right]\] which are equal the dimensions of \[k\].

Hence, the correct option is C.

Note: In the above solution, we have taken the dimensions of the constant k equal to the dimensions of pressure. But one can also determine the dimensions of k by calculating the dimensions of any of terms kinetic energy or potential energy in Bernoulli's equation as they all have the same dimensions.