Appendix A. Fundamentals of Science

Homework Problem Solving  1. List the “given” variables with their symbols, values and units.  2. List the unknown variables to “Find” with units, symbols, etc.  3. Draw a sketch of the situation.  4. Determine which equation(s) contains the unknown variables as a function of the knowns.  If the solution equation contains more unknowns, find additional equations for them.

 5. Make assumptions, if necessary, for any unknowns for which you have no equations. Clearly state your assumptions and verify them.  6. Solve the equations using the known or assumed values, being certain to carry along the units. Show your intermediate steps.  7. Identify the final answer by putting a box around it.

 8. Check your answer. Is it logical. If the solved units don’t match the desired units of the unknown, then either a mistake was made, or unit conversion might be needed. Also, certain functions such as “ln” and “exp” require arguments that are dimensionless, while trig functions, like “sine” need an argument in degrees or radians.

7 basic dimensions, 2 supplementary dimensions.

Supplementary dimensions

 Notice Tables A-3 (Derived Dimensinos) and A-4 (Prefixes)

Unit Conversion  Multiplying a quantity in one set of units by the ratio of equivalent units does not change the value of the quantity. 

 Suppose you don’t know the equivalent of 1 unit in another set of units. E.g. suppose you don’t know how much 1 lb/in 2 is in units of millibars.  If you know the value of average sea level pressure in each set of units, their ratio is still 1, so: 

 This works as long as each set of units has the same zero point. If they have different zero points, an adjustment must be made.  Fahrenheit and Celsius degrees:  1 o C = 1.8 o F  Convert 12 o C to Fahrenheit. By the previous method, we get.  But, since they have different zero points (freezing is at 0 o C, but at 32 o F) we have to account for the difference in zero starting point.

 So, we add 32 degrees to the answer.  To go from o F to o C, we must first subtract 32 o and then use the ratio method.  To go from o C to o K, just add o

Functions and Finite Difference  Independent variable: When considering two values, the value of the independent variable does not depend on the value of the other variable.  Dependent variable: The value of the dependent variable does depends on the value of the other variable.  Note: the value of a dependent variable can depend on several independent variables.

  means “a change in.”  Independent variable is time.  Independent variable is height  Change must be in the same direction as the independent variable.  Gradient: Change of something with distance. May include, wind speed, pressure, moisture content, etc.

Relationships and Graphs  Convention is to have independent variable on abscissa (horizontal axis) and the dependent on the ordinate (vertical axis).  In meteorology, the ordinate is often taken as the independent variable, such as height, and the abscissa as the dependent variable.

Linear, Logrithmic, Exponential  Relationships which are linear form a straight line when plotted on linear graph.  Relationships which are logarithmic or exponential form straight lines when plotted on semi-log graphs. (One axis linear and one axis logarithmic).  Relationships in which the dependent variable is proportional to the power of the independent variable will form a straight line on log-log graphs.

 One can get a good idea of the equation which fits a set a data points simply by the type of graph on which the plotted data produces a straight line.

Semi-Log Paper  Consider the exponential relationship. When plotted on semi-log paper, it produces a straight line.

 This is a log-log graph.  If we were to plot the equation: we would get a line that looks like the next plot.

 If we take the logarithm of both sides we get: in which we can see is the equation for a straight line of y-intercept log a, and slope of b. From the graph, y-intercept = 2.5

 Problems: Units Matrix handout, N1, U4 (note: N1 is asking for the conversion values. U4 is asking for the answers to be in the basic SI units).  SHOW ALL EQUATIONS USED AND CALCULATIONS