Fluid Mechanics (College Board AP® Physics 1: Algebra-Based): Exam Questions

Figure 1

Students perform an experiment with a cylinder which is filled with water, as shown in Figure 1. The students make a small hole in the side of the cylinder and measure the speed at which water exits the hole. The students plug the first hole, make another one at a different height, and repeat this procedure. Table 1 shows the height of each hole relative to the top of the water, and the corresponding water speed .

Table 1

The students correctly determine that the relationship between and is given by . The students create a graph with plotted on the vertical axis.

i) Indicate which measured or calculated quantity could be plotted on the horizontal axis to yield a linear graph whose slope can be used to calculate an experimental value for the acceleration due to gravity. Use the blank columns in the table to list any calculated quantities you will graph other than the data provided.

Vertical axis: Horizontal axis:

ii) On the grid in Figure 2, plot the appropriate quantities to determine the acceleration due to gravity. Clearly scale and label all axes, including units, as appropriate.

Figure 2

iii) Draw a best fit line for the data graphed in part ii).

Calculate an experimental value for the acceleration due to gravity using the best-fit line that you drew in part a)iii).

Figure 1

A syringe contains an ideal fluid. The plunger of a syringe is pushed and fluid leaves the needle of the syringe. The barrel has cross sectional area and the inside of the needle has cross sectional area .

The plunger exerts force on the fluid.

Identify force experienced by the fluid in the needle in terms of . Justify your answer using Newton's third law applied to fluids.

A large boat has a mass , and can displace a maximum volume . The boat is floating in a river with water of density and is being loaded with steel beams each of density and volume . The boat owners want to be able to carry as many beams as possible.

Derive an expression for the maximum number of steel beams that can be loaded on the boat without exceeding the maximum displaced volume, in terms of the given quantities and physical constants, as appropriate.

A fluid flows through a tube. At a given point, the tube has a cross-sectional area . Fluid passes this point with a speed .

Derive an expression for the volume of fluid passing the given point per unit time. Give your expression in terms of , and physical constants as appropriate.

Figure 1

Tank Y is filled with water up to a height ​, as shown in Figure 1. The top of the tank is open to the atmosphere, and the bottom is connected to a short horizontal pipe of radius . A valve at the bottom is initially closed but is opened at time , allowing water to flow out.

The surface of the water in the tank moves downward at a constant speed ​ as the water exits the pipe with velocity ​. The tank is designed so that the cross-sectional radius of the top water surface changes as the water level drops.

i) Starting from Bernoulli’s equation, derive an expression for the speed ​ at which water exits the pipe in terms of and .

ii) Starting from the continuity equation, derive a relationship between the exit velocity ​ of the water and the radius of the top water surface. Write this relationship in terms of , and physical constants as appropriate.

iii) When is much larger than , it is often assumed that:

​

Justify why this approximation is valid using the expressions derived in (i) and (ii).