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Chapter 4: Problem 70

In a laboratory experiment, the water flow rate is to be measured catching the water as it vertically exits a pipe into an empty open tank that is on a zeroed balance. The tank is \(10 \mathrm{m}\) directly below the pipe exit, and the pipe diameter is \(50 \mathrm{mm} .\) One student obtains a flow rate by noting that after 60 s the volume of water (at \(4^{\circ} \mathrm{C}\) ) in the tank was \(2 \mathrm{m}^{3}\). Another student obtains a flow rate by reading the instantaneous weight accumulated of \(3150 \mathrm{kg}\) indicated at the \(60-\mathrm{s}\) point. Find the mass flow rate each student computes. Why do they disagree? Which one is more accurate? Show that the magnitude of the discrepancy can be explained by any concept you may have.

Short Answer

Expert verified
The first student's computed mass flow rate is \(33.33 \, kg/s\) and the second student's computed mass flow rate is \(5.35 \, kg/s\). The discrepancy is due to the second student measuring the weight of the water, not its mass, which has resulted in an overestimate of the mass flow rate.

Step by step solution

01

Compute the mass flow rate for the first student

The first student measured the volume of water after 60 seconds to be \(2 \, m^{3}\). The mass of this volume of water can be calculated using the density of water at \(4 ^{\circ}C\), which is \(1000 \, kg/m^{3}\). So, the mass is \(1000 \, kg/m^{3} * 2 \, m^{3} = 2000 \, kg\). The mass flow rate is then this mass divided by the time it took to accumulate, i.e., \(2000 \, kg/60 \, s = 33.33 \, kg/s\).
02

Compute the mass flow rate for the second student

The second student measured the accumulated weight at 60 seconds to be \(3150 \, kg\). As weight is mass times gravity (\(9.81 m/s^{2}\)), we need to divide this weight by gravity to get the mass: \(3150 \, kg / 9.81 \, m/s^{2} = 321.1 \, kg\). Then, by dividing this mass by the time it took to accumulate, we get the mass flow rate: \(321.1 \, kg / 60 \, s = 5.35 \, kg/s\).
03

Analyze the discrepancy

The mass flow rates computed by the two students are significantly different because the second student measured the weight of the water, not its mass. The weight is subject to the gravitational strength at the point of measurement, which in this case is at an elevation of 10 meters, and therefore slightly less than at sea level where the standard \(9.81 \, m/s^{2}\) applies. This has led to a slight overestimate of the mass and thereby the mass flow rate by the second student.

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