Lab 1
Fundamental Laboratory Skills
Objective: To provide the fundamental laboratory skills necessary for analyzing experimental data.
Theory: The following sections provide the necessary background information required to complete this laboratory.
I.Uncertainty and Significant Figures
A.Every measurement has some uncertainty in the value.
Example: We measure the height of a basketball goal using a ruler. We find that the height of the goal is
H = ( 120. 0.5 ) inches
This method of writing a measurement is called interval form It says that the actual value lies
between 119.5 inches and 120.5 inches.
The value of 120.inches is your best guess at the height and is also called the central value.
The value of 0.5inches is called the absolute uncertainty in the measurement.
Reducing absolute uncertainty in a measurement costs money and takes skill.
B.Although the absolute uncertainty in a measurement is an important quantity, it doesn't provide us with a means to compare two measurements involving two different objects. For example, measuring Mount Everest's height to an absolute uncertainty of 0.5 inches is a much better measurement job than measuring the length of a grasshopper to 0.5 inches.
C.The relative uncertainty of a measurement is defined as the absolute uncertainty of a measurement divided by the central value.
Example: The relative uncertainty in the height of the basketball goal measurement is given by
D.The measurement with lowest relative uncertainty is the most precise measurement (covered later).
E.Scientists and engineers need some way to indicate the uncertainty (tolerance) in their design parameters. They use significant digits. (see Chapter 1 of textbook for more details)
Example: h = 120. inches implies (120 0.5) inches
h = 120 inches implies (120 5) inches
h = 120.0 inches implies (120 0.05) inches
h = 1.20x103 inches implies (1,200 5) inches
II.Accuracy and Precision
A.Precision has to do with the spreadin the values of repeatedmeasurements while accuracy has to do with how far your best guess is off from the correct value.
Example:Four different ROTC students shoot four shots at a target as shown below:
The first shooter had good precision (small spread) but had poor accuracy (all of the shots were high and left of the target).
The second shooter had poor precision (large spread) but had good accuracy (all of the shots are grouped around the bulls-eye of the target).
The third shooter had poor precision (large spread) and poor accuracy (to the right of the target).
The last shooter had good precision (small spread) and good accuracy (all of the shots hit the bulls-eye.
Example 2: Given that the correct value of a physical quantity is 10. meters and that the experimental measurements by four students are shown below
#1: ( 9.0 0.1 ) m
#2 : ( 10.0 0.6 ) m
#3 : (115 8 ) m
#4 : (10.00 0.07) m
which of the measurements are precise and which are accurate.
Solution: Graph each of the measurement intervals on a number line. If the correct value lies within the measurement interval then the measurement is accurate otherwise it is inaccurate.
The width of the measurement interval tells you if your measurement is precise. The smaller the measurement interval then the more precise the measurement.
Prove to yourself that the results are the same as for the previous example.
III.Random and Systematic Errors
A.All measurements have some error. The possible types of errors are random and systematic.
B.Random errorsincrease the spread of a series of measurements while systematic errorsshift the measurements. Thus, random errors effect precision while systematic errors effect accuracy. When analyzing an experiment, it is important to determine the primary type of error so that you could reduce it if the experiment was repeated.
Note:In all of our discussions, we have stated that we know the correct value of a particular physical quantity! How do we know this value? One possibility is that a particular theory predicts this value. But, if this value is only predicted by theory, how do we know if our experiment suffers from systematic error or if the theory is wrong. The answer is that our results must be reproducible by scientists around the world and that there may be other types of experiments for measuring this physical quantity. This enables us to compare our results in order to determine if the theory or experiment is correct.
IV.Statistics and Measurement
A.One way to reduce random errors is to measure the same object several times. In this manner, we hope that the random errors will sum up to zero. However, if we have several values (guess') for the object then how do we choose our best guess?
Answer: If we several measurements, we can use statistics. We choose the average (mean) of our measurements as our best guess. We use the standard deviation of the measurements as our absolute uncertainty. For N measurements of the physical quantity x, we have that
Experimental Mean
Standard Deviation
Example: The following measurements in meters were made for the height of hill, write the confidence interval for the measurement:
{ 110., 111, 108, 113, 107, 111, 118, 105, 107, 110., 112, 109, 109, 111, 109, 108 }
Solution: and
Thus our final answer is ( 110. 2 ) meters
B.The reason that the denominator of the standard deviation is reduced to N-1 is that the freedom of one measurement is already lost when we calculated the mean. Stated another way, if we were told the mean value of a given set of measurements then after N-1 values are written down, we already know the last measurement's value!
C.As the number of identical measurements approach infinity, the discrete measurements approach a the continuous Gaussian distribution (Bell curve). This greatly simplifies the calculation of the standard deviation. This is often the case for physics and engineering experiments where 1022 atoms may be involved.
Percent (Relative) Error For A Gaussian
Absolute Uncertainty
IV.Graphing Skills
A.Linear - Linear Graph
1.This type of graph plots one variable on the y-axis against a second variable on the x-axis. You should have used this type of graph in Algebra class.
2.Its popularity is due to the physiological makeup of humans. Our eyes enable us to quickly determine if the graph of a set of data approximately forms a straight line. Using a ruler, we can quickly make a best guess of the slope, m, and y-intercept, b. You can also make guess' on the maximum and minimum values of the slope and y-intercept.
y = m x + b
3.We can see no other shape as easily as a straight line!! Thus, the other graph types are tricks to get straight lines for other functions!
B.Semilog Graph
1.In this graph, the logarithm of one variable is plotted along the y-axis against another variable
on the x-axis.
2.If experimental data shows an exponential dependence then the graph will be a straight line on this type of graph paper.
Proof: Consider the function where we plot the Log(N) on the y-axis and t on the x-axis
3.The slope of your line gives you k!
4.The y-intercept of the line gives you the initial value A!
5.Calculating the slope of a line on semi-log paper is the same as for lin-lin graph paper
Hint: When using semi-log paper choose your two points where the functions has risen an integer number of intervals. The number of intervals risen by the line is the numerator in your slope calculations so you don't need a Calculator. Isaac Newton didn't have a Calculator or Maple to solve these problems.
C.Log-Log Graph
1.In this graph the logarithm of one variable is plotted on the y-axis and the logarithm of the othervariable is plotted on the x-axis. (You must use the same base for both logarithms!)
2.If experimental data shows a power relationship then the graph will be a straight lineon this type of graph paper.
Proof: Consider the function where we plot the Log(F) on the y-axis and Log(v) on the x-axis
3.The slope of your line gives you the power, k!
4.The y-intercept of the line gives you the initial value A!
5.Calculating the slope of a line on log-log paper is the same as for lin-lin graph paper
Hint: When using Log-Log paper choose your two points where the functions where both rise and run are an integer number of intervals. The number of intervals risen by the line is the numerator and the number of intervals for the run is the denominator in your slope calculations so you don't need a Calculator.
Experimental Procedure:
I.Measurement
1.During the first part of lab, you will make a measurement at each one of four different stations:
StationA - Measure the length of an object using a ruler.
Station B - Measure the length of an object using a vernier caliper
Station C - Measure the mass of an object using a balance
Station D - Measure the time it takes a pendulum to make 5 complete cycles.
2.You will record your measurement from each station onto the blackboard in the correct column.
3.As soon as everyone in the lab has posted their measurements, you will use the data to determine the confidence intervals for all four measurements. Write both the 67% confidence levels and 90% confidence levels.
67%:(best guess +/- s)90%: (best guess +/- 2s)
II.Graphing Data
1.Manually plot each of the following sets of data until you obtain a straight-line graph on one of the different types of graph papers.
2.Use your graph to obtain the functions that best represent the data.
3.Replot your data using Excel and use linear regression to obtain the functional form.
x / y / x / y / x / y0.01 / 0 / 1.0 / 5.04 / 5 / 126
0.2 / 0.22 / 1.5 / 11.22 / 8 / 321
0.5 / 1.25 / 2 / 20.1 / 10 / 496
x / y / x / y / x / y
0.01 / 0.1 / 1.0 / 119.5 / 5 / 602
0.2 / 24.0 / 1.5 / 181 / 8 / 956
0.5 / 60.5 / 2 / 239 / 10 / 1203
x / y / x / y / x / y
0.01 / 0.310 / 1.0 / 4.8 / 3 / 1,196
0.2 / 0.53 / 1.5 / 19.0 / 4 / 18,920
0.5 / 1.18 / 2 / 75.0 / 5 / 305,000