AP Physics 1 Unit 8 Review: Fluids

Draw shaded bars that represent $P_1$, $\tfrac12\rho v_1^2$, and $\rho g y_1$ to complete the bar chart in Figure 2 for point 1. Figure 3 shows a bar chart representing the three terms in Bernoulli's equation (each term is energy per unit volume) at point 2: the pressure term $P$, the kinetic term $\tfrac12\rho v^2$, and the gravitational term $\rho g y$. Points 1 and 2 are at the same height, and the pressure at point 2 is $P_{\text{atm}}$ because the nozzle discharges to the atmosphere.

Starting with a fundamental physics principle or an equation from the reference information, derive an expression for the exit speed $v_2$ of the water at point 2. Express your answer in terms of $P_{\text{air}}$, $P_{\text{atm}}$, $\rho$, $g$, and $h$ only. Because the nozzle discharges to the atmosphere, $P_2 = P_{\text{atm}}$. Use mass conservation and energy conservation for steady incompressible flow between the tank free surface and point 2. The speed of the water at the free surface is negligible because the tank is large.

Indicate whether the magnitude of the force the student must exert on the plate to hold it in place is greater than, less than, or equal to the force that would be required if the tank were open to the atmosphere (so $P_{\text{air}} = P_{\text{atm}}$). A student temporarily covers the nozzle exit with a flat circular plate of diameter $d = 2.0\ \text{cm}$, as shown in Figure 5, while the tank conditions remain the same and the flow is brought to rest at the exit. Assume the water at the exit is momentarily stationary so the pressure on the inside face of the plate equals the static pressure of the water at point 2 when the exit is blocked. Take $P_{\text{air}} = 160\ \text{kPa}$, $P_{\text{atm}} = 101\ \text{kPa}$, $\rho = 1000\ \text{kg/m}^3$, $g = 9.8\ \text{m/s}^2$, and $h = 0.80\ \text{m}$.

Indicate whether the magnitude of the tension in the string is greater than, less than, or equal to the magnitude of the buoyant force on the sphere. The spherical object is held fully submerged and at rest relative to the pipe by the string. The water exerts a buoyant force on the sphere. The water also exerts a pressure force on all points of the sphere's surface, but because the sphere is small compared with the pipe diameter at section 1, the pressure in the surrounding water at the sphere's location may be treated as approximately uniform at a given height.

Refer to Figure 1. Indicate whether the pressure at location 2 in Scenario 1, $P_2$, is greater than, less than, or equal to the pressure at location 2 in Scenario 2, $P_2'$, by writing one of the following in your answer booklet.

Starting with a fundamental physics principle for fluids and the mass conservation relationship, derive an expression for the pressure difference $P_1 - P_2$. Express your answer in terms of $\rho$, $Q$, $A_1$, and $A_2$ only. Begin your derivation by writing a fundamental physics principle or an equation from the reference information. Consider the general case of steady, incompressible, nonviscous flow through a horizontal pipe from location 1 (area $A_1$, pressure $P_1$) to location 2 (area $A_2$, pressure $P_2$) with constant volumetric flow rate $Q$ and fluid density $\rho$.

Indicate whether the expression for $P_1 - P_2$ you derived in part B is or is not consistent with the claim made in part A. Briefly justify your answer by referencing your derivation in part B and how changing the area at location 2 affects the pressure at location 2.