Fluid Mechanics

Young's modulus is given by: where: is the cross-sectional area, and is the force. Since the stress is the force divided by area, an increase in area does not affect the Young's modulus, meaning there is no change to the Young's modulus. Hence, the change is .

The buoyant force is given by the formula: where: (density of ice), (volume displaced), (acceleration due to gravity). Substituting the values:

The pressure change depends on the flow rate and other factors like the velocity of the fluid, which cannot be determined just from the change in radius. Therefore, more information is needed to determine the new pressure.

For a liquid drop floating in another liquid, the buoyancy force is balanced by the surface tension. The relationship between the diameter of the drop and the surface tension is given by:

The force exerted on the wall is equal to the rate of change of momentum of the water hitting the wall.
The momentum flux is given by: where is the mass flow rate and is the velocity of the water.
The mass flow rate is: where is the density of water, is the cross-sectional area of the tube, and is the velocity. Substituting the values: Thus, the force is: Therefore, the correct answer is (a).

The liquid will not come out of the hole when the horizontal acceleration of the vessel causes the liquid to remain stationary relative to the vessel.
This condition can be met if the effective acceleration due to gravity in the horizontal direction is balanced by the force due to the liquid column's height. The horizontal acceleration required can be found using the following relation derived from the balance of forces:
Thus, the correct answer is (d).

The viscous force on a body moving in a fluid is given by the Stokes' law: where is the viscosity of the fluid, is the radius of the body, and is the velocity of the body through the fluid. In this case, the fluid is glycerine with density , and the force is proportional to the mass of the body and the difference in densities.
Thus, the correct answer is (c).

The rate of flow of liquid out of the hole is proportional to the square root of the height of liquid, as given by Torricelli's law.
Therefore, the time required to empty the liquid is inversely proportional to the square root of the height.
Let the total time to empty the tank be .
The time to empty the first half of the tank is given by: The time to empty the second half of the tank is: The ratio of to is:
Thus, the correct answer is (d).

The Reynolds number is given by the equation:

where is the density of the fluid, is the velocity of the fluid, is the diameter of the pipe, and is the dynamic viscosity of the fluid.

As seen from the formula, the Reynolds number depends on the velocity of the fluid, the diameter of the pipe, and the viscosity of the fluid, but it is independent of the length of the pipe.

Hence, the correct answer is (c).

The terminal velocity of a liquid drop is proportional to the square of its radius , as given by: When two drops of equal radii coalesce, their combined radius becomes . The new terminal velocity will be proportional to the square of the new radius: Thus, the new terminal velocity will be . Hence, the correct answer is (c).