Test: Bernoulli’s Principle - JEE MCQ

Bernoulli’s theorem, in fluid dynamics, relation among the pressure, velocity, and elevation in a moving fluid (liquid or gas), the compressibility and viscosity (internal friction) of which are negligible and the flow of which is steady, or laminar. First derived (1738) by the Swiss mathematician Daniel Bernoulli, the theorem states, in effect, that the total mechanical energy of the flowing fluid, comprising the energy associated with fluid pressure, the gravitational potential energy of elevation, and the kinetic energy of fluid motion, remains constant. Bernoulli’s theorem is the principle of energy conservation for ideal fluids in steady, or streamline, flow and is the basis for many engineering applications.

Bernoulli's theorem, in fluid dynamics, relation among the pressure, velocity, and elevation in a moving fluid (liquid or gas), the compressibility and viscosity (internal friction) of which are negligible and the flow of which is steady, or laminar.

We know that Pv remains constant for any fluid and for non compressible fluids like water we get that Pv is always constant.
Also as 76cm of Hg = 10⁵ Pa  
We get 2m Hg = 200/76 x 10⁵ Pa
Thus from conservation of Pv, we get that
2 x  200/76 x 10⁵  = 4 x P
Thus we get P = 100/76 x 10⁵
Thus we get P = 1.31 x 10⁵

Applying Bernoulli's principle, we have
P₁+1/2​ρv₁²=P₂+1/2​ρv₂² 
⇒P₂​−P₁=1/2ρ (v2/1−v2/2)
⇒ΔP=1/2​×1.3× (1202−90²)
⇒ΔP=0.65× (120+90) × (120−90)
⇒ΔP=0.65×210×30=4095Pa

Bernoulli's principle is applicable on those non-viscous liquids which have laminar flow or streamlined flow. It means that a liquid in which its particles exert no force on each other, i.e., the speed of all the particles of the liquid is same.

Also, the liquid should travel in the form of streamlines, i.e., the liquid flowing in one pipeline (imaginary pipeline) should not mix in the liquid in another pipeline. This is called streamlined flow.

Also, turbulent flow is the opposite of streamlined flow. So, turbulent liquid will not obey Bernoulli's principle. But if the liquid is rotated, this streamlined flow will not take place. Thus Bernoulli's principle will not be applicable to this type of liquid.

 

Every foot of elevation change causes a 0.433 PSI change in water pressure. If your pipe is going downhill add 0.433 PSI of pressure per vertical foot the pipe goes down. If the pipe is going uphill subtract 0.433 PSI for every vertical foot the pipe goes up.

We know that the continuity theorem says that if the cross sectional area of the water flow decreases, the speed must increase to maintain the volume of water flown. And according to Bernoulli's principle if the speed of water flow increases , then the pressure must decrease.

In streamline flow, the product of cross section area and velocity remains constant (equation of continuity). So in the narrowest part of the pipe velocity is maximum. 
And from Bernoulli's theorem, we know that the sum of potential energy, kinetic energy and pressure energy remains constant. Since pipe is horizontal potential energy is equal at all the points. So the narrowest part  of pipe pressure (pressure energy) will be minimum because velocity (kinetic energy) is maximum in the narrowest part.

Bernoulli's theorem-Bernoulli's theorem, in fluid dynamics, relation among the pressure, velocity, and elevation in a moving fluid (liquid or gas), the compressibility and viscosity (internal friction) of which are negligible and the flow of which is steady, or laminar.

Bernoulli's principle can even be derived from the principle of conservation of energy. Bernoulli's principle states that, in a steady flow, the sum of all forms of energy in a fluid along a streamline is the same at all points on that streamline.
Hence D is the correct answer.