Homework #7
Conceptual Solution

These solutions rely on the extensive use of two equations:

1. (b). Using the continuity equation we find that:

   

   Because (as given in the passage), is double . Because (as given in the passage), must be equal to . This means that is the greatest.

2. (c). Because the area is being cut in half as the fluid flows from Region 1 into Region 2, the velocity of the fluid must be increasing, ultimately by a factor of 2 (the relative sizes of the two regions). This eliminates choices (a) and (d). When the velocity increases, the term in Equation 2 must increase, so for the overall equation to stay constant, the pressure must be reduced. The exact amount by which the pressure decreases is uncertain, because and are not given. This makes (c) the best choice.
3. (a). We can see from Bernoulli's equation that the two factors affecting the pressure that vary between the regions are the height (y) and the velocity (v). The following relationship holds true:

   Because , . This means that is the greatest velocity. As given in the passage, . In order for the equality to hold true, must be greater than both and . The greatest pressure is found in Region 1, where the area is greatest and the tube is lowest. The best answer is choice (a).

4. (d). As the fluid flows from Region 1 to Region 2, the cross sectional area decreases, so the velocity must increase according to the continuity equation. Because the term in Bernoulli's equation (Equation 2) increases while the term is constant, the pressure must decrease. This eliminates choices (a) and (c). Choices (b) and (d) differ in their comparison only of the pressures in Regions 1 and 3. Because their cross-sectional areas are the same, both regions have the same flow velocity. Because Region 3 is higher than Region 1, Region 3 must have a lower pressure than Region 1 if Bernoulli's equation is to remain balanced. Choice (d) is correct. (Note that this question is a little buggy; you don't really have enough information to know that , as claimed in choice (d).)
5. (a.) Using the continuity equation, if increases, then must decrease. If decreases while remains unchanged, Bernoulli's equation says the pressure must increase. Increasing (the height of Region 2) will reduce the pressure in that region, so choice (c) should be eliminated. Changing should have no direct effect on the pressure in Region 2. The best answer is choice (a).
6. (d). Comparing the pressures between two different regions requires employing Bernoulli's equation. Since , the continuity equation implies . Bernoulli's equation thus reduces to:

   

   To make equal to , must be equal to (given that ). For this to occur, must increase, or must decrease. This makes statements I and III valid. The lengthening of Region 2 will have no effect on the pressures in Region 1 or Region 3, so the best answer is choice (d).

7. (c). According to the continuity equation:

   

   If decreases, must increase to satisfy the continuity equation. This makes choice (c) the best answer. According to Bernoulli's equation, should decrease if increases-which rules out choice (a). Velocity and pressure in Region 3 will have no bearing on the pressure in Region 2. This makes choices (b) and (d) invalid.

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