AP Physics 1 Unit 8 Review: Fluids

Draw shaded bars that represent $P_{\text{atm}}$, $P_g$, and $P_{\text{total}}$ to complete the pressure bar charts in Figure 2 and Figure 3 for cross section 1 and cross section 2, respectively. Figure 4 shows a bar chart that represents the atmospheric pressure $P_{\text{atm}}$, the gauge pressure $P_g$, and the total pressure $P_{\text{total}}$ of the water just under the piston. Gauge pressure is defined by $P_g = P_{\text{total}} - P_{\text{atm}}$. Atmospheric pressure is the same at all locations. The bars in Figure 4 establish the scale for pressure.

Starting with a fundamental physics principle, derive an equation for the total pressure $P_{2}$ at cross section 2 in terms of $m_p$, $A_p$, $h$, $\rho$, $g$, and $P_{\text{atm}}$. Your derivation must include (1) an equation representing the force balance on the piston and (2) an equation relating the pressure at cross section 2 to the pressure just under the piston. Begin your derivation by writing a fundamental physics principle or an equation from the reference information. The piston rises slowly, so the piston is in vertical force equilibrium at all times (see Figure 5). The bottom of the piston is $h = 0.60\ \text{m}$ above cross section 2. The centers of cross sections 1 and 2 are at the same height. The piston has $m_p = 3.0\ \text{kg}$ and $A_p = 2.0× 10^{-3}\ \text{m}^2$. Use $\rho = 1000\ \text{kg/m}^3$, $g = 9.8\ \text{m/s}^2$, and $P_{\text{atm}} = 1.01× 10^5\ \text{Pa}$.

Indicate whether the speed $v_2$ of the water at cross section 2 is greater than, less than, or equal to the speed $v_1$ at cross section 1. Use $r_1 = 2.0\ \text{cm}$, $r_2 = 1.0\ \text{cm}$, and $v_1 = 1.5\ \text{m/s}$ at cross section 1. Water is incompressible.

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.