Physics is all about measuring, calculating and graphing.  In chemistry, we had simple rules for rounding answers to proper significant digits and our graphs were always linear.  Those were the good old days!  

If the uncertainty on a measurement is larger than expected, the old rounding rules from chemistry may not give us correct precision.  In this lab, we are going to delve into the details of how to record valid measurements based on the measuring device used and how to perform calculations with those measured values.  In addition, we'll look at how to graph data when it's not linear.

Part A: Measurement and Error Analysis

Every measurement you make must be recorded with an uncertainty value.  The uncertainty is often ± 1 on the final digit, but not always.  The value you use will depend on the measuring device's scale and your ability to read it.  Small, tightly packed scales, jostling, and unclear data like a ragged surface could all increase the uncertainty of the measurement.  

When adding and subtracting, absolute uncertainty values add directly.  Thus, 

	(2.0 ± 0.1)  +  (3.0 ± 0.2)  =  5.0 ± 0.3

When multiplying and dividing, you need to add the relative errors instead.  Recall that relative error is equal to the absolute error divided by the measurement.  Therefore, if you want to multiply the values above, you'll first need to convert to RE.

	0.1 / 2.0 = 0.05	and		0.2 / 3.0 = 0.07

Multiplying the measurements and adding RE values gives us 

	(2.0 ± 0.1)  *  (3.0 ± 0.2)  =  6.0 with a Re of 0.12

Convert back to absolute error by taking the measurement times AE, 6.0 * 0.12 = 0.7.  So our final answer is

	(2.0 ± 0.1)  *  (3.0 ± 0.2)  =  6.0 ± 0.7

Methodology

0.	Skip down to part B, methodology step 1, and do that before you start.  That way it will be ready for you when you get there!

1.	Measure the length, width, and height of a textbook.  Use centimeters as your unit and be sure to record the values with proper uncertainties.

2.	Imagine lining three identical books along the edge of your lab desk.  The first is placed so the length of the book lines up with the edge of the desk, the second is turned so that its width lines up along the edge, and the third is turned so that its height lines up with the edge.  Calculate the total distance the three books would cover along the edge of the desk and express your answer with proper precision and include uncertainty values.

3.	Calculate the surface area of the cover of the book with proper uncertainty.

4.	Calculate the volume of the book with proper uncertainty.

5.	Find the thickness of a single page of the book four different ways.  Use millimeters as your unit and (of course) attach uncertainty to the measurements.  First, measure the thickness of one page directly.  Next, measure the thickness of 50 pages and divide the value by 50 to get the average thickness.  You'll have to calculate the uncertainty on your answer, remembering that 50 is a counting number with no uncertainty.  Then do the same thing with 100 pages and with 200 pages.

Analysis

Most of the analysis is already covered in the methodology section.  Make sure you perform all calculations to proper significant figures and that you perform error analysis along with each calculation.  Show all work clearly.

Part B: Logarithmic Graphing

It's nice when graphs turn out linear.  That happened a lot in chemistry--Charles' law, densities, Beer's Law, and more all produced linear graphs.  But what if the results are not linear?  You can make nonlinear graphs, of course, but it is more difficult to pull useful information from them.  For example, if you look at a graph of y = x2, y = x3, and y = x4 for fairly small positive values of x, you'll see the graphs are very different to distinguish from each other, especially if you add in random lab error.

The way to fix this problem is to find a way to make the graph linear.  How?  Logs to the rescue!  Consider the general equation

	y = C xn

where C is some constant and n is the exponent on x.  If we were to take the log of both sides, and remembering that multiplication becomes addition for logs, we get

	log y = log C + log xn

But of course exponents turn into multipliers inside a log, so moving the exponent n in front of the x term and doing some slight rearranging, we get

	Log y = n log x + log C

But notice that's just a linear equation!  So if on the x-axis we graph log x and on the y-axis we graph log y, we should get a straight line whose slope is n and whose y-intercept is log C.  This is a convenient way to find an unknown exponent, by the way, since it's just the slope of the line.

In this lab, we're going to explore logarithmic graphing by experimenting with Newton's Law of Cooling.  Yeah, he did everything!  Imagine you have a nice hot cup of Earl Grey, but before you can drink it, you have to get some work done first.  The tea will cool as time goes on, of course, until it reaches the temperature of the air in the room (which we call ambient temperature).  But how fast does it cool?  At first, it cools rapidly because it's temperature is very different from ambient, but as time goes on and it gets cooler, the rate of cooling decreases.  In other words, the rate of cooling depends on the difference in temperatures.

Processes like that end up as exponential functions whose general form is y = C e-kx, where C and k are constants and e is Euler's number.  In our case, the y value will be the difference  between the tea's current temperature and ambient temperature, and the x value will be the time that has passed.  So we have

	T - Ta = (To - Ta) e-kt

Where T is the current temperature of the tea, To is the original temperature of the tea, Ta is ambient temperature, t is time, and k is a constant for the system.  In ths case we'll want to take the natural long, ln, rather than just log.  Remember that ln e = 1, so we'll end up with

	ln (T - Ta) = -kt + ln (To - Ta)

That is a linear equation, so if we graph ln (T - Ta) on the y-axis and t on the x-axis, we'll get a linear graph.  Note that in this case only one axis is graphed logarithmically while the other is graphed normally.  We call this a semilogarithmic graph.

Methodology

1.	Pour approximately 100 mL of water into a beaker and heat it up to somewhere just under the boiling point.  Somewhere around 80oC or 90oC will be fine.  You might want to get this started at the beginning of the period so you don't have to wait for it to heat up.

2.	Using tongs, remove the beaker from the hotplate and immediately measure the water's temperature.  This is To.  

3.	At 30 second intervals, record the temperature of the water.  When you reach a point where the temperature is changing by very small amounts, start recording at 60 second intervals instead.  As time goes on and the rate of cooling decreases, you'll want to keep switching to longer intervals, 120 s (2 min), 5 min, and so on.  You're done when the temperature doesn't change appreciably in a 5 minute interval.

4.	At some point, don't forget to measure the temperature of the air in the room.  This is ambient temperature, Ta.

Analysis

You'll want to set up a data table in a spreadsheet that includes columns for time t, temperature T, T-Ta, and ln (T-Ta).  Make sure you express all times in seconds.  Construct a graph the traditional way, with T-Ta on the y-axis and t on the x-axis.  You should see an exponential decay graph, which is curved and therefore of limited use.  Now graph semilogarithmically, with ln (T - Ta) on the y-axis and t on the x-axis.  Now the graph should be linear.  The slope will be equal to -k and the y-intercept will be equal to ln (To - Ta).  Since you know both To and Ta, you can calculate that value and see if it agrees with the intercept of your graph.

Questions

1.	Which had more uncertainty, the calculated area or the calculated volume?  Why?

2.	Did your measured page thicknesses agree with each other within the range of uncertainty?  Why or why not?

3.	Which page thickness had the smallest uncertainty?  Which had the largest?  Why?

4.	What was the value of graphing temperatures on a logarithmic scale?

5.	The period of a pendulum (T) is given by the equation T = 2p √(L/9.8) where L is the length of the pendulum.  Show that if you take the log of both sides the graph will be linear.  What would you graph on each axis?  What will the slope of the line be?  What will the y-intercept be?