Fluid Mechanics

Step 1: Relation for SHM.
For a particle undergoing simple harmonic motion (SHM), the potential energy and restoring force are related by the equation: and where is the spring constant.
Step 2: Relating , , and .
By substituting into the equation for , we get:
Step 3: Conclusion.
Thus, the correct answer is (B) .

Step 1: Applying Bernoulli’s principle.
When the valve is opened, the pressure energy of the liquid is converted into kinetic energy of flow.
Step 2: Pressure difference and velocity relation.
The pressure drop is According to Bernoulli’s equation,
Step 3: Solving for velocity.

Step 4: Conclusion.
The speed of water flowing out is

Step 1: Continuity equation.
For incompressible fluid:
Thus, velocity is maximum where cross-sectional area is minimum.

Step 2: Bernoulli’s theorem.
For horizontal flow:

Step 3: Pressure–velocity relation.
Increase in velocity causes decrease in pressure.

Step 4: Conclusion.
At the narrowest part of the pipe, velocity is maximum and pressure is minimum.

Step 1: Apply equation of continuity.


Step 2: Write areas in terms of diameters.


Step 3: Solve for nozzle diameter.

Step 1: Write Torricelli’s theorem.
Velocity of efflux at depth :
Step 2: Write discharge formula.

Step 3: Write discharge from both holes.
For hole A (depth , area ):
For hole B (depth , area ):
Step 4: Equate discharges.

Step 5: Solve for .

Step 6: Conclusion.
The side of square hole is .

Step 1: Understanding the effect of depth on the capillary rise.
The capillary rise in a tube depends on the surface tension, which is influenced by the depth of the tube below the surface of the water. As the tube is taken to a depth inside the mine, the effective height of water rise changes due to the increase in atmospheric pressure with depth. Step 2: Deriving the relation.
At a depth , the water level rises less due to the increased pressure, which is directly proportional to the height of the water. Therefore, the ratio of the heights at the surface and at depth is: Step 3: Conclusion.
Thus, the correct answer is (B) .

Step 1: Concept
For an adiabatic process, .
Step 2: Analysis
Volume .
So, .
Step 3: Calculation
For a monoatomic gas, .
.
Thus, .
Step 4: Conclusion
The correct ratio is .
Final Answer: (C)

Concept:
Potential energy of a dipole in uniform electric field: Step 1: Initial position. Dipole is parallel to field:
Step 2: Final position. Dipole rotated by :
Step 3: Energy required.
Step 4: Conclusion. Energy required =