Measurements, Units and Mathematical Equations in Physics

Equations in physics relate the results of measurements. A measurement in turn is simply a comparison of two related quantities. For example I might report to you that the distance from here to New York City is about three times the distance from here to Washington DC. I have reported the result of my measurement of the distance to New York in units of the distance to DC. Now while this result is useful for someone in the Delmarva area it is quite useless for someone without a clue of the JHU-DC distance. So for the purpose of being able to accurately communicate the result of measurements a system of units has been agreed upon. Note however that even though we use these standard units in the end we are only making a comparison between quantities whenever we perform a measurement.

In mechanics the relevant physical properties we deal with are length, mass and time. The units which we shall use for these quantities are meters, kilogram and seconds, that is we shall be using the SI also called the MKS unit system. As you can imagine precision and reproducibility is crucial and you can read in the book about how this has been achieved through the appropriate definitions of each of these basic units.

Rather than spend time on the exact definitions used in this unit system I want to give you an important recommendation concerning the use of units which will help you solve problems correctly.

1. Assign a symbol to every physical quantity relevant in your problem.
2. Manipulate Physics equations with these symbols leaving numbers out until you have a final symbolic expression for the
3. When you put in numbers to equations always carry the units along with numbers.

A simple example of this: A typical heart pumps blood at a rate of 0.083 l/s. How long does it take for the 1.25 Gallons of blood to pass through the body? We denote S=0.083 l/s the pumping speed, V=1.3 Gallons the total volume of blood, and t is the time to be determined. We first derive a symbolic equation for the time, t:

Looking at this formula we clearly see that the time decreases if the volume of blood, V, increases of the pumping speed decreases. This is as expected and indicates that we have gotten the formula right. Then we put in numbers with their corresponding units

This techniques will help you think clearly about the physics by examining the symbolic expression and catch certain mistakes which reveal themselves by producing the wrong units for the result when you put in numbers with their units.