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Chapter 2: Problem 1

Using dimensional equations, convert (a) 2 wk to microseconds. (b) \(38.1 \mathrm{ft} / \mathrm{s}\) to kilometers/h. (c) \(554 \mathrm{m}^{4} /\) (day \(\cdot \mathrm{kg}\) ) to \(\mathrm{ft}^{4} /\left(\min \cdot \mathrm{lb}_{\mathrm{m}}\right)\)

Short Answer

Expert verified
The converted units are (a) approximately \(1.21 \times 10^{14}\) microseconds; (b) approximately \(41.15 \, \text{km/hr}\); (c) approximately \(3.91 \times 10^{3} \, \text{ft}^4 \, \text{lbm} / \text{min}\).

Step by step solution

01

Convert 2 weeks to microseconds

Note that 1 week = 7 days; 1 day = 24 hours; 1 hour = 60 minutes; 1 minute = 60 seconds; 1 second = \(1 \times 10^{6}\) microseconds. Use this chain of equivalences to convert 2 weeks to microseconds by multiplying \(2 \cdot 7 \cdot 24 \cdot 60 \cdot 60 \cdot 10^6\).
02

Convert \(38.1 \, \text{ft/s}\) to \(\text{km/hr}\)

Using the conversion factors 1 km = 3280.8 ft, 1 hour = 3600 seconds; convert \(38.1 \, \text{ft/s}\) to \(\text{km/hr}\) by multiplying \((38.1 \, \text{ft/s}) \cdot \frac{1 \, \text{km}}{3280.8 \, \text{ft}} \cdot \frac{3600 \, \text{s}}{1 \, \text{hr}}\).
03

Convert \(554 \, \text{m}^4\) / \(\text{day} \cdot \text{kg}\) to \(\text{ft}^4\) / \(\text{min} \cdot \text{lbm}\)

Utilize the following conversions: \(1 \, \text{m} = 3.2808 \, \text{ft}\); \(1 \, \text{day} = 1440 \, \text{min}\); \(1 \, \text{kg} = 2.2046 \, \text{lbm}\). Convert \(554 \, \text{m}^4\) / \(\text{day} \cdot \text{kg}\) to \(\text{ft}^4\) / \(\text{min} \cdot \text{lbm}\) as: \(554 \, \text{m}^4\) / \(\text{day} \cdot \text{kg}\) \(\times \frac{(3.2808 \, \text{ft})^4}{1 \, \text{m}^4}\) \(\times \frac{1 \, \text{day}}{1440 \, \text{min}}\) \(\times \frac{1 \, \text{kg}}{2.2046 \, \text{lbm}}\)

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Key Concepts

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

Dimensional Analysis
Dimensional analysis is a method used to convert between different units by using the relationships between them. It helps in checking the consistency of equations and calculations by ensuring that the dimensions on both sides match. This process is crucial when we want to convert units as seen in various scientific and mathematical tasks.

This method involves:
  • Identifying the original unit and the unit you wish to convert to.
  • Using conversion factors that equate different units to each other.
  • Multiplying by these conversion factors until the desired unit is achieved.
Dimensional analysis serves not only as an aid in unit conversion but also in verifying the physical correctness of scientific equations. When performing conversions, it鈥檚 important to apply conversion factors appropriately, ensuring that all the given quantities maintain the same overall dimension.
SI Units
The International System of Units (SI) is the modern form of the metric system. It's the most widely used system of measurement around the world for science and commerce. SI units provide a standard, which makes scientific communication and experimentation more straightforward and uniform.

The main SI units include:
  • Meter (m) for length
  • Kilogram (kg) for mass
  • Second (s) for time
While converting to or from SI units, understanding the basic units of measure is crucial. For example, converting time from weeks to microseconds or distance from feet to meters or kilometers involves knowing how many smaller units (like seconds or feet) fit into larger SI units (like microseconds or kilometers). This understanding forms the basis for dimensional analysis using these standards.
Conversion Factors
Conversion factors are critical bridges in dimensional analysis. A conversion factor is a ratio that expresses how many of one unit are equal to another unit. This ratio can be used to multiply a quantity in order to convert it from one unit to another without changing its value.

For example:
  • To convert time from weeks to microseconds: 1 week equals 604800 seconds, and 1 second equals 1,000,000 microseconds.
  • To convert speed from feet per second to kilometers per hour: 1 kilometer equals 3280.8 feet, and 1 hour equals 3600 seconds.
  • To convert units of measurement from meters to feet: 1 meter equals 3.2808 feet.
Conversion factors are generally derived from known equivalences. Consistent application of these ratios ensures accurate and meaningful results in calculations and helps maintain the integrity of scientific measurements.

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Most popular questions from this chapter

Sketch the plots described below and calculate the equations for \(y(x)\) from the given information. The plots are all straight lines. Note that the given coordinates refer to abscissa and ordinate values, not \(x\) and \(y\) values. [The solution of Part (a) is given as an example.] (a) A plot of In \(y\) versus \(x\) on rectangular coordinates passes through \((1.0,0.693)\) and \((2.0,0.0)\) (i.e., at the first point \(x=1.0\) and \(\ln y=0.693\) ). (b) A semilog plot of \(y\) (logarithmic axis) versus \(x\) passes through (1,2) and (2,1). (c) A log plot of \(y\) versus \(x\) passes through (1,2) and (2,1). (d) A semilog plot of \(x y\) (logarithmic axis) versus \(y / x\) passes through (1.0,40.2) and (2.0,807.0). (e) A log plot of \(y^{2} / x\) versus \((x-2)\) passes through (1.0,40.2) and (2.0,807.0).

A right circular cone of base radius \(R\), height \(H\), and known density \(\rho_{s}\) floats base down in a liquid of unknown density \(\rho_{f}\). A height \(h\) of the cone is above the liquid surface. Derive a formula for \(\rho_{f}\) in terms of \(\rho_{s}, R,\) and \(h / H,\) simplifying it algebraically to the greatest possible extent. [Recall Archimedes' principle, stated in the preceding problem, and note that the volume of a cone equals (base area)(height)/3.]

The relationship between the pressure \(P\) and volume \(V\) of the air in a cylinder during the upstroke of a piston in an air compressor can be expressed as \(P V^{k}=C\) where \(k\) and \(C\) are constants. During a compression test, the following data are taken: $$\begin{array}{|c|c|c|c|c|c|c|}\hline P(\mathrm{mm} \mathrm{Hg}) & 760 & 1140 & 1520 & 2280 & 3040 & 3800 \\ \hline V\left(\mathrm{cm}^{3}\right) & 48.3 & 37.4 & 31.3 & 24.1 & 20.0 & 17.4 \\\\\hline\end{array}$$ Determine the values of \(k\) and \(C\) that best fit the data. (Give both numerical values and units.)

On a website devoted to answering engineering questions, viewers were invited to determine how much power a 100 -MW power plant generates annually. The answer declared to be best was submitted by a civil engineering student, who stated, "It produces \(100 \mathrm{MW} / \mathrm{hr}\) so over the year that's \(100^{*} 24^{*} 365.25 \&\) do the math." (a) Carry out the calculation, showing all the units. (b) What is wrong with the statement of the question? (c) Why was the student wrong in saying that the plant produces \(100 \mathrm{MW} / \mathrm{hr} ?\)

L-Serine is an amino acid important for its roles in synthesizing other amino acids and for its use in intravenous feeding solutions. It is often synthesized commercially by fermentation, and recovered by subjecting the fermentation broth to several processing steps and then crystallizing the serine from an aqueous solution. The solubilities of L-serine (L-Ser) in water have been measured at several temperatures, producing the following data: \(^{5}\). $$\begin{array}{|c|c|c|c|c|c|c|c|c|}\hline T(\mathrm{K}) & 283.4 & 285.9 & 289.3 & 299.1 & 316.0 & 317.8 & 322.9 & 327.1 \\ \hline x \text { (mole fraction L-Ser) } & 0.0400 & 0.0426 & 0.0523 & 0.0702 & 0.1091 & 0.1144 & 0.1181 & 0.1248 \\ \hline\end{array}$$ One of the ways such data can be represented is with the van't Hoff equation: \(\ln x=(a / T)+b\) Graph the data so that the resulting plot is linear. Estimate \(a\) and \(b\) and give their units.
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