## MCAT Physics and Math Review

**Chapter 4: Fluids**

### Practice Questions

1. Objects A and B are submerged at a depth of 1 m in a liquid with a specific gravity of 0.877. Given that the density of object B is one-third that of object A and that the gauge pressure of object A is 3 atm, what is the gauge pressure of object B? (Note: Assume atmospheric pressure is 1 atm and )

1. 1 atm

2. 2 atm

3. 3 atm

4. 9 atm

2. An anchor made of iron weighs 833 N on the deck of a ship. If the anchor is now suspended in seawater by a massless chain, what is the tension in the chain? (Note: The density of iron is and the density of seawater is )

1. 100 N

2. 724 N

3. 833 N

4. 957 N

3. Two wooden balls of equal volume but different density are held beneath the surface of a container of water. Ball A has a density of and ball B has a density of When the balls are released, they will accelerate upward to the surface. What is the relationship between the acceleration of ball A and that of ball B?

1. Ball A has the greater acceleration.

2. Ball B has the greater acceleration.

3. Balls A and B have the same acceleration.

4. It cannot be determined from information given.

4. Water flows from a pipe of diameter 0.15 m into one of diameter 0.2 m. If the speed in the 0.15 m pipe is what is the speed in the 0.2 m pipe?

1.

2.

3.

4.

5. A hydraulic lever is used to lift a heavy hospital bed, requiring an amount of work *W*. When the same bed with a patient is lifted, the work required is doubled. How can the cross-sectional area of the platform on which the bed is lifted be changed so that the pressure on the hydraulic lever remains constant?

1. The cross-sectional area must be doubled.

2. The cross-sectional area must be halved.

3. The cross-sectional area must be divided by four.

4. The cross-sectional area must remain constant.

6. The figure shown represents a section through a horizontal pipe of varying diameters into which four open vertical pipes connect. If water is allowed to flow through the pipe in the direction indicated, in which of the vertical pipes will the water level be lowest?

1. Pipe 1

2. Pipe 2

3. Pipe 3

4. Pipe 4

7. The speed of blood in the aorta is much higher than the speed of blood through a capillary bed. How can this fact be explained using the continuity equation, assuming that we are interested in average flow and that there is no net fluid loss?

1. The aorta is located higher than the capillary bed.

2. The pressure in the aorta is the same as the pressure in the capillary bed.

3. The cross-sectional area of all the capillaries added together is much greater than the cross-sectional area of the aorta.

4. The cross-sectional area of a capillary is much smaller than the cross-sectional area of the aorta.

8. Which of the following data sets is sufficient to determine the linear speed through an area of a rigid pipe?

1. The cross sectional area in another segment of pipe and the cross sectional area in the region of interest

2. The Reynolds number, viscosity of the fluid, density, and diameter of the pipe

3. The radius of the pipe, pressure gradient, viscosity, and length of the pipe

4. The absolute pressure and density

9. A large cylinder is filled with equal volumes of two immiscible fluids. A balloon is submerged in the first fluid; the gauge pressure in the balloon at the deepest point in the first fluid is found to be 3 atm. Next, the balloon is lowered all the way to the bottom of the second fluid, where the hydrostatic pressure in the balloon reads 8 atm. What is the ratio of the gauge pressure accounted for by the first fluid to the gauge pressure accounted for by the second fluid?

1. 1:3

2. 3:4

3. 3:5

4. 3:8

10.An hydraulic system is designed to allow water levels to change depending on a force applied at the top of the tank as shown. If a force, *F*_{1}, of 4 N is applied to a square, flexible cover where *A*_{1} = 16, and the area *A*_{2} = 64, what force must be applied to *A*_{2} to keep the water levels from changing?

1. 4 N

2. 16 N

3. 32 N

4. No force needs to be applied.

11.Balls A and B of equal mass are floating in a swimming pool, as shown below. Which will produce a greater buoyant force?

1. Ball A

2. Ball B

3. The forces will be equal.

4. It is impossible to know without knowing the volume of each ball.

12.Bernoulli’s equation is the reason for the upward force that permits airplane flight. Which statement best summarizes the equation’s relationship to flight?

1. The speed of airflow is equal on the top and bottom of a wing, resulting in nonturbulent flight.

2. The speed of airflow is greater over the curved top of the wing, resulting in less pressure on the top of the wing and the production of a net upward force on the wing, in turn resulting in flight.

3. The speed of airflow on the flat bottom of the wing is greater than over the curved top of the wing, resulting in more pressure below the wing and the production of a net upward force on the wing, in turn resulting in flight.

4. The weight of the wing is directly proportional to the weight of the air it displaces.

13.A low-pressure weather system can decrease the atmospheric pressure from 1 atm to 0.99 atm. By what percent will this decrease the force on a rectangular window from the outside? (Note: Assume the window is 6 m by 3 m and the glass is 3 cm thick.)

1. 1%

2. 10%

3.

4. 30%

14.Two fluids, A and B, have densities of *x* and 2*x*, respectively. They are tested independently to assess absolute pressure at varying depths. At what depths will the pressure below the surface of these two fluids be equal?

1. Whenever the depth of fluid A is one-half that of fluid B

2. Whenever the depth of fluid A equals that of fluid B

3. Whenever the depth of fluid A is 2 times that of fluid B

4. Whenever the depth of fluid A is 4 times that of fluid B

15.A water tower operator is interested in increasing the pressure of a column of water that is applied to a piston. She hopes that increasing the pressure will increase the force being applied to the piston. The only way to increase the pressure is to alter the speed of the water as it flows through the pipe to the piston. How should the speed of the water be changed to increase the pressure and force?

1. Increase the speed

2. Decrease the speed

3. Release water intermittently against the pipe

4. The speed of water will not change pressure at the piston.

PRACTICE QUESTIONS

### Answers and Explanations

1. **C**The absolute and gauge pressures depend only on the density of the fluid, not that of the object. When the pressure at the surface is equal to atmospheric pressure, the gauge pressure is given by *P*_{gauge} = *ρ*g*z*, where *ρ* represents the density of the fluid, not the object. These objects are also at the same depth, so they must have the same gauge pressure.

2. **B**

The tension in the chain is the difference between the anchor’s weight and the buoyant force because the object is in translational equilibrium: **T** = **F**_{g} – **F**_{buoy}. The object’s weight is 833 N, and the buoyant force can be found using Archimedes’ principle. The magnitude of the buoyant force is equal to the weight of the seawater that the anchor displaces:

*F*_{buoy} = *ρ*_{w}*V*_{w}g

Because the anchor is submerged entirely, the volume of the water displaced is equal to the volume of the anchor, which is equal to its mass divided by its density We are not given the anchor’s mass, but its value must be the magnitude of the weight of the anchor divided by g. Putting all of this together, we can obtain the buoyant force:

Lastly, we can obtain the tension from the initial equation **T** = **F**_{g} – **F**_{buoy}:

**T** = 833 N − 109 N = 724 N

3. **A**

Using Newton’s second law, *F*_{net} = *ma*, we obtain the following equation:

*F*_{buoy} − *m*g = *ma*

Thus,

Both balls experience the same buoyant force because they are in the same liquid and have the same volume (*F*_{buoy} = *ρV*g). Thus, the ball with the smaller mass experiences the greater acceleration. Because both balls have the same volume, the ball with the smaller density has the smaller mass (*m* = *ρV*), which is ball A.

4. **B**

It is known that water flows faster through a narrower pipe. The speed is inversely proportional to the cross-sectional area of the pipe because the same volume of water must pass by each point at each time interval. Let A be the 0.15 m pipe and B the 0.20 m pipe, and use the continuity equation:

*ν*_{A}*A*_{A} = *ν*_{B}*A*_{B}

where *ν* is the speed and *A* is the cross-sectional area of the pipe. Because *ν* is inversely proportional to the cross-sectional area, and the area is proportional to the square of the diameter we obtain the following:

**Choice (B)** most closely matches our result and is thus the correct answer.

5. **A**This question tests our understanding of Pascal’s principle, which states that a change in pressure applied to an enclosed fluid is transmitted undiminished to every portion of the fluid and to the walls of the containing vessel. We are told that the work required to lift the bed with the patient is double the work needed to lift just the bed. In other words, the force required doubles when both the bed and the patient have to be lifted. To maintain the same pressure, we must double the cross-sectional area of the platform of the hydraulic lever on which the patient and the bed are lifted.

6. **B**It is not necessary to do any calculations to answer this question. The open vertical pipes are exposed to the same atmospheric pressure; therefore, differences in the heights of the columns of water in the vertical pipes is dependent only on the differences in hydrostatic pressures in the horizontal pipe. Because the horizontal pipe has variable cross-sectional area, water will flow the fastest and the hydrostatic pressure will have its lowest value where the horizontal pipe is narrowest; this is called the Venturi effect. As a result, pipe 2 will have the lowest water level.

7. **C**The continuity equation states that the flow rate of a fluid must remain constant from one cross-section to another. In other words, when an ideal fluid flows from a pipe with a large cross-sectional area to one that is narrower, its speed decreases. This can be illustrated through the equation *A*_{1}*ν*_{1} = *A*_{2}*ν*_{2}. If blood flows much more slowly through the capillaries, we can infer that the cross-sectional area is larger. This might seem surprising at first glance, but given that each blood vessel divides into thousands of little capillaries, it is not hard to imagine that adding the cross-sectional areas of each capillary from an entire capillary bed results in an area that is larger than the cross-sectional area of the aorta.

8. **C**The data given in **choice (C)** are sufficient to determine the flow rate through Poiseuille’s law, which can then be used to determine the linear speed by dividing by the cross-sectional area (which could be determined from the radius, as well). **Choice (A)** would be sufficient if we also knew the flow rate in the other segment of pipe; one could use the continuity equation to determine the linear speed. The data in **choice (B)** could be used to determine the critical speed at which turbulent flow begins, but there is no indication that there is turbulent flow. The data in **choice (D)** could be used to determine the depth of an object in a fluid.

9. **B**The first step in answering this question is defining the different types of pressures. Atmospheric pressure is the pressure at the top of the first fluid exerted by air (at sea level, it is equal to 1 atm). Gauge pressure is the pressure inside the balloon above and beyond atmospheric pressure; gauge pressure is the total (absolute or hydrostatic) pressure inside the balloon minus the atmospheric pressure. Gauge pressure depends on the density of the fluid, the constant of gravity, and the depth at which the object is submerged. Hydrostatic or absolute pressure is the total pressure in the balloon (that is, the gauge pressure and the atmospheric pressure together). Because we are given the gauge pressure at the bottom of the first fluid as 3 atm, our task now is to calculate the gauge pressure accounted for by the second fluid. The hydrostatic pressure at the bottom of the cylinder is 8 atm. One of these atmospheres is atmospheric pressure pushing on the fluids. Another 3 atmospheres are accounted for by the first fluid that is pushing on the second fluid. Thus, the gauge pressure due to the second fluid is 8 − 1 − 3 = 4atm. The ratio of the gauge pressures is therefore 3:4.

10.**B**This is a basic restatement of Pascal’s principle that a force applied to an area will be transmitted through a fluid. This will result in changing fluid levels through the system. The relationship is stated as Plugging in the numbers gives an answer of 16 N.

11.**A**

The buoyant force (*F*_{buoy}) is equal to the weight of water displaced, which is quantitatively expressed as

*F*_{buoy} = *m*_{fluid displaced}g = *ρ*_{fluid}*V*_{fluid displaced}g

The volume of displaced fluid is equal to the volume of the ball. The density of the fluid remains constant. Therefore, because ball A has a larger volume, it will displace more water and experience a larger buoyant force.

12.**B**Airplane wings have curved upper surfaces and flat lower surfaces, which causes the air to flow faster over the top of the wing because it has further to travel to the edge of the wing than the air over the flat bottom surface. Increased air speed will mean lower pressure within the fluid. This will result in higher pressure below the wing and an upward force.

13.**A**This question is a simple application of the definition of pressure, which is force per area. If pressure decreases 1 percent and area does not change, the force will be decreased by 1 percent. Note that the other measurements given do not play a role in our calculations.

14.**C**

The equation for absolute (hydrostatic) pressure is *P* = *P*_{o} + *ρ*g*z*, where *P*_{o} is the pressure at the surface, *ρ* is the density of the fluid, g is acceleration due to gravity, and *z* is the depth in the fluid. If the density of fluid B is twice that of fluid A, then the depth in fluid A will have to be twice that in fluid B to obtain the same absolute pressure:

15.**B**This is a basic interpretation of Bernoulli’s equation that states, at equal heights, speed and pressure of a fluid are inversely related (the Venturi effect). Decreasing the speed of the water will therefore increase its pressure. An increase in pressure over a given area will result in increased force being transmitted to the piston.