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 $v$ 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 $h$ of each hole relative to the top of the water, and the corresponding water speed $v$.

Table 1

The students correctly determine that the relationship between $h$ and $v$ is given by

$v=\sqrt{2gh}$

The students want to determine an experimental value for the acceleration due to gravity. The students create a graph with $v^2$ plotted on the vertical axis.

i) Label the horizontal axis of Figure 2 with a measured or calculated quantity. Include units, as appropriate. The graphed quantities should yield a linear graph that can be used to determine the acceleration due to gravity.

ii) On the grid provided in Figure 2, create a graph of the quantities indicated in part a)i).

• Clearly label the horizontal axis with a numerical scale

• Plot the corresponding data points on the grid

• Any columns added to Table 1 for scratch work will not be scored.

Figure 2

iii) Draw a best fit line for the data plotted in part a)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 $A$ and the inside of the needle has cross-sectional area $0.05A$.

The plunger exerts force $F$ on the fluid.

Determine the force experienced by the fluid in the needle in terms of $F$. Justify your answer.

A large boat has a mass $M_b$, and can displace a maximum volume $V_b$. The boat is floating in a river with water of density ${\rho }_w$ and is being loaded with steel beams each of density ${\rho }_s$and volume $V_s$. The boat owners want to be able to carry as many beams as possible.

Derive an expression for the maximum number $N$ 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 $A$. Fluid passes this point with a speed $v$.

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

Figure 1

Tank Y is filled with water up to a height $h_0$​, 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 $r$. A valve at the bottom is initially closed but is opened at time $t=0\mathrm{s}$, allowing water to flow out.

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

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

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

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

​$v_p\approx \sqrt{2gh}$

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