91Ó°ÊÓ

Perform the following estimations without using a calculator. (a) Estimate the mass of water (kg) in an Olympic-size swimming pool. (b) A drinking glass is being filled from a pitcher. Estimate the mass flow rate of the water (g/s). (c) Twelve male heavyweight boxers coincidentally get on the same elevator in Great Britain. Posted on the elevator wall is a sign that gives the maximum safe combined weight of the passengers, \(W_{\mathrm{max}},\) in stones. (A stone is a unit of mass equal to \(14 \mathrm{lb}_{\mathrm{m}}\). It is commonly used in England as a measure of body weight, which, like the numerical equivalence between the \(1 \mathrm{b}_{\mathrm{m}}\) and \(\mathrm{Ib}_{\mathrm{f}},\) is only valid at or near sea level.) If you were one of the boxers, estimate the lowest value of \(W_{\max }\) for which you would feel comfortable remaining on the elevator. (d) The Trans-Alaska Pipeline has an outside diameter of 4 ft and extends 800 miles from the North Slope of Alaska to the northernmost ice-free port in Valdez, Alaska. How many barrels of oil are required to fill the pipeline? (e) Estimate the volume of your body \(\left(\mathrm{cm}^{3}\right)\) in two different ways. (Show your work.) (f) A solid block is dropped into water and very slowly sinks to the bottom. Estimate its specific gravity.

Step by step solution

01

(a) Estimating Mass of Water in an Olympic Pool

Firstly, the volume of an Olympic swimming pool is approximately \(50 \, m \times 25 \, m \times 2 \, m = 2500 \, m^3\). Water has a density of about \(1000 \, kg/m^3\), so the total mass is \(2500 \, m^3 \times 1000 \, kg/m^3 = 2,500,000 \, kg\).

02

(b) Estimating the Mass Flow Rate

From observation, about two full glasses can be poured from the pitcher in one second and each glass might hold quarter of a litre (0.25 \(kg\)) of water, therefore, the mass flow rate of the water would be \((0.25 \, kg \times 2) / 1 \, s = 0.5 \, kg/s = 500 \, g/s\).

03

(c) Determining Safe Weight in Stones

An average male heavyweight boxer weigh around 100 kg, or 220 lbs, which is roughly 15 stones. Therefore, for 12 boxers, the total weight would be 12 x 15 = 180 stones.\nFor reasonable safety, let's account for a 20% margin, so \(W_{\mathrm{max}}\) should be at least \(1.2 \times 180 = 216 \) stones.

04

(d) Calculating Oil Volume for Pipeline

The volume of the pipeline can be determined by the formula for the volume of a cylinder \(V = \pi d h\). Given that diameter (\(d\)) is 4 ft and height (\(h\)) is 800 miles (approximately 4,224,000 ft), the volume is \(\pi \times 2^2 \times 4,224,000 = 106,150,000 ft^3\). As 1 barrel of oil is approximately \(0.158987 \, m^3 = 5.615 \, ft^3\), the number of barrels required to fill the pipeline is \(106,150,000 \, ft^3 / 5.615 \, ft^3/barrel = 18,901,669\) barrels of oil.

05

(e) Estimating Body Volume

First, using the water displacement method, fill a bathtub to the brim and get in. The water overflowing is your body volume. As an adult, approximate the volume as about 70 litres, or 70,000 \(cm^3\).\nSecond, if we approximate the human body as a cylinder, the volume could be estimated as: \(V = \pi r^2 h\). Average adult height (\(h\)) is about 170 cms and radius (\(r\)) might be 15 cms (waist radius). Substituting the values, \(V = \pi \times 15^2 \times 170 = 120,530 \, cm^3\).

06

(f) Estimating Specific Gravity

Considering it sinks very slowly, the block is slightly denser than water. The specific gravity, ratio of density of the block to the density of water, will still be approximately 1.0 (a bit more), but its exact value will depend on the exact density of the block, which can vary depending on material.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Mass and Volume Calculations

Understanding mass and volume calculations is crucial for several fields including physics, chemistry, and engineering. Mass refers to the amount of matter in an object, typically measured in grams or kilograms, while volume is the space that an object occupies, measured in liters or cubic meters. To estimate the mass of water in an Olympic-size swimming pool, one must know the water's volume and density. Water has a standard density of approximately 1 kg/l or 1000 kg/m³. By multiplying this density by the pool's volume, you can estimate the mass of water it contains.

Estimating mass flow rate involves calculating how much mass passes through a given point per unit time. Observations and reasonable assumptions play a critical role; for example, imagining how many glasses can be poured per second and the volume (hence mass) of each glass. Units must be consistent when executing these calculations to avoid errors, which means often converting from liters to kilograms or from seconds to hours, depending on the context.

Units of Measurement Conversion

Units of measurement conversion is essential to ensure that calculations are accurate and meaningful. It's the process of converting one unit of measurement to another. In an international context, it is important to be familiar with both the Metric system (liters, meters, kilograms) and the Imperial system (gallons, feet, pounds).

For example, in Great Britain, mass is sometimes measured in stones, which must be converted to pounds (14 lb/st) or kilograms when necessary. To ensure safety in situations like the elevator scenario posed in the exercise, you would convert the boxers' weight into stones and account for a safety margin. This concept illustrates the real-world importance of being able to skillfully convert between units of measurement, which can range from everyday situations to complex scientific computations.

Specific Gravity Estimation

Specific gravity estimation illustrates how you can use the concept of density to make inferences about an object. Specific gravity is a dimensionless quantity that compares the density of a substance to the density of a reference substance, typically water for liquids and solids. The specific gravity of water is by definition 1. If an object sinks slowly in water, as described in the exercise, it means its specific gravity is slightly greater than 1. An estimation doesn't require precise measurement but instead relies on a qualitative analysis and comparisons. This is a basic, yet valuable method in material science and engineering to quickly assess whether materials will float or sink in a given fluid, which can influence design decisions in applications ranging from shipbuilding to the manufacture of flotation devices.