Welcome to the mechanics laboratory!
Today we will spend some time discussing the contents, requirements, and procedures of this course as seen in the syllabus . The syllabus can be found under my lab main page, http://www2.hawaii.edu/~jmcfatri/labs/mechlab.html .  In the website you will find many useful pages, including the schedule, your grades, Java applets, physics links, and all the notes for the course, including these notes.

After discussing the syllabus and after briefly covering reports , we will go over Chapter 2 of the General Physics Laboratory I: Mechanics lab book. You should go back and read the section on "writing reports" in my website, as I ask students to follow a strict format.

Chapter 2
pp 7 - 12 of your lab book

In this section, we will cover basic laboratory skills.  I assume this is a review for all of you, and I will cover it very quickly. The basic skills we will discuss are:

• Units and Dimensional Analysis
• Creating Tables and Graphs
• Linearizing equations

Units and Dimensional Analysis
It is important that every physical quantity you measure is given with its proper unit. If you do not include the unit, or provide the wrong units, confusion can arise. To demonstrate this, we will play a quick game I call "The Unit Jumble." Play it on the web at http://www2.hawaii.edu/~jmcfatri/Java/UnitJumble.html .

Dimensional analysis is a method of analyzing equations by looking at the units (also known as dimensions) of the quantities involved. It can be very useful because it allows you to

• check if the form of your equation is right
• check to see if you have made any algebraic mistakes in a series of equations

Performing a dimensional analysis is quite easy.  All you have to do is remember a few simple, common sense rules.  These rules are:

1. Units on each side of an equal sign must be the same (unless the equation is a conversion factor).
2. Terms in a summation or subtraction must have the same units.
3. An exponent must be unitless.

Let's use these rules to figure out which of these equations is correct:

1. Distance = velocity / time
2. Distance = velocity * time
3. Distance = velocity
4. Distance = velocity * time²

The left hand side (LHS) has dimensions of length.  If the equation has the correct form then the right hand side (RHS) must also have a dimension of length. So let's look at the dimensions of the RHS of all four. Velocity has units of [length] / [time] (e.g. mph). You can treat dimensions just as if they were algebraic quantities.

	RHS in terms of dimensions	Result
1	[length]     1   [time]  [time]	[length] [time]2
2	[length]  [time] [time]	[length]
3	[length] [time]	[length] [time]
4	[length]  [time]2 [time]	[length][time]

Since only the second equation has the correct units, it is the only one of those 4 equations which could be correct.  (Note however, that this method says nothing of constants!  According to our analysis, 2 * velocity * time would work equally well!)

Tables and Graphs
I will assume you know how to construct a table and display a graph.  If not, you can see samples of both under "writing reports" as well as in your lab book.  A brief example of both will be presented in class.

Linearizing Equations
In this lab, we will often be performing a task called linearizing equations.  When you linearize an equation, you are putting that equation into the form y = mx + b.  The use of this, I suppose, was to be able to see quickly whether or not your equation fits the data. You could plot y versus x by hand and see on your graph if the plot was linear.  This is something of an "old school" method of doing things.  Today, it is often more instructive to leave the equation in its original form, plot raw data, and use a computer program to see how well your expression fits the data. However, since we must do this in the lab, I will cover it.

Suppose we have an equation:

A(T) = A ₀ e^-E/2kT

If you tried to plot data values of A(T) versus T, you would not get a line.  Suppose your goal in the lab was to find E.  It is simple enough to fit your A(T) versus T data to this equation and leave A₀ and E as free parameters which a fitting program could calculate. However, you could also linearize the equation by taking the natural log of both sides of the equation.

ln(A(T)) = ln(A₀) - E/2kT
or
ln (A(T)) = - (E/2k) * (1/T) + ln(A₀)
 

This is the same form as y = mx + b, if y = ln (A(T)), m = -E/2k, x = 1/T, and b = ln (A₀).  Unfortunately, you have measured A(T) and T directly.  In order to plot this function, you must change all of your data values.  You have to take the inverse of all your T values (to get 1/T) and you have to take the log of all your A(T) values.  If you change your values in this way the plot of ln (A(T) versus (1/T) will appear linear if your equation is correct.

The downfall of this approach, besides giving you extra work to do, is that oftentimes we don't know what the function should be.  Using a fitting program which can do arbitrary fits, allows us to see easily what function fits best. We have software in this lab which can do arbitrary fits (GA by Vernier Software).  The same software also does linear fits.