Measurement errors
Measurements are associated with errors or uncertainties and for that reason we report only the appropriate number of significant digits. For example, if our uncertainty in a quantity is in the hundredths place such as ±0.05 m, there is no reason to report the thousandths place so we would round 8.377 m to 8.38 m. Errors could be either random or systematic. If you measure the length of a pencil as a single piece of data three times you may read or record three slightly different answers. Each measurement is independent of the other and the error associated with that is called a random error. If the meterstick has an error in its calibration that affects all the data, this is called a systematic error. These errors affect the final outcome of the experiment.
Linear relationships - Straight line graphs
In experiments we investigate how two or more quantities are related to each other. If two quantities are linearly dependent on each other, then the relationship between the two is linear and can be represented by a straight line. Since it is familiar to us, let's consider two variables, x and y, that are linearly related. Where x is the independent variable and y is the dependent variable. The equation representing the relationship is written as
y = mx + b
where the slope is m and the intercept is b. If we measure data for x and y, then plot y versus x, we should see a linear trend for the data. If x and y are normally distributed random variables, then we can use a statistical procedure to calculate the slope and the intercept of the equation. The relevant statistical procedure is called the Least Squares Method (LSM). The LSM provides ways to calculate both the value and its error or uncertainty for slope and uncertainty.
Part B: Experiment
In our first experiment we will use length measurements to calculate the value of π. You will calculate the value of π with its uncertainty and compare with the known standard value.
C = 2πR where C is circumference and R is radius.
Predictions
1. If we measure the circumference and radius of a variety of circular objects and then plot circumference vs. radius, will the data fall exactly on a straight line? Why or why not?
No. This is because the radius and circumference have two different measurements. The radius is the distance from the circle to its perimeter, and the circumference is the distance once around the circle. Thus, the data for the radius and circumference will be different plotted on a line graph.
2. What do you expect the slope of the line to be equal to? Explain why.
3. What do you expect the intercept of the line to be equal to? Explain why.