Introduction to Measurement and Error Analysis.

 

 

The Goal

You have to learn how to make the measurements of one of the basic physical quantity - length. You will also get an introduction to measurement errors associated with these measurements.

 

Prerequisites

“Physics for Scientists and Engineers” R.D. Knight: Chapters 1.9.

 

Equipment

-         Cardboard box

-         Meter Stick

           

Procedure

     You primary goal is to calculate the volume of the cardboard box. To make calculations you have to measure box’s width, depth, and height. The volume can then be determine as a product

 

V = width x depth x height

 

However the box is not perfect. For example, its width is slightly different depending at which place you are doing your measurement. To ensure more reliable data make at least five measurements of each quantity at different places. Do not measure one variable (width, for example) five times consequently to obtain more reliable results. Fill in appropriate columns of tables (for the width the table is shown below) using SI standard units and numbers in scientific notation.

 

Width

Error of measurement

2.37 x 101 m

1.0 x 10-2 m

 

 

 

 

 

 

 

 

    

Note, that experimental uncertainties (errors of measurements) always exist and your prime function, as a scientist, to minimize them. Try to answer the following questions:

-         What is the source of errors during single measurement of the length?

-         What factor does determine the value of that error?

-         Can we minimize that error? What do we have to do for that?

 

Determine the minimal values of your errors of measurements and fill in the right columns in appropriate tables. Note, that you have no ways to minimize these errors – they are simply determined by the procedure and tools used in the process of measurement. These limitations on the precision of a measurement are commonly called scale errors or scale uncertainties.

Scale uncertainty is a limiting factor in the precision of a measurement. It is due to the fact that a measuring scale can have only finitely many divisions. It is reasonable (and expected) to estimate one digit between the finest markings on the scale (if possible).

                                          

 

Analysis of data

1.    Examine the data in left column of the “Width” table. Note, that particular numbers are slightly different due to imperfections of the box. To estimate the average width of the box calculate the mean value of the width (just arithmetic average) as

                                                           

     where n is the number of measurements. This value gives as rather good  

     approximation of the overall cardboard box width.

2.    Now estimate how far away from this value our particular results of measurements are with the help of standard deviation σ

                                                   

 

     Standard deviation allows us to estimate uncertainty interval for the mean

     value of a set of measurements in the form

                                                  

 

3.  Now you have two different estimates for the errors:

     - scale uncertainty

     -  

   

     To make final decision about errors you can simply pick up the largest

      one.

3.    Repeat previous steps for the length and depth.

4.    Calculate the volume of the cardboard box using the mean values for width, length, and depth.

5.    Estimate the error of the volume using the following simple rules:

·        When adding or subtracting measurements, add their absolute uncertainties. Symbolically, if the most probable value of the sum of two measurements x and y is s (s = x + y), and if Δx and Δy are the absolute uncertainties of x and y respectively, then the uncertainty in s, Δs, is: Δs= Δx+ Δy

·        When multiplying or dividing measurements, add their relative uncertainties. Symbolically, if x and y are two measurements, and if their absolute uncertainties are Δx and Δy, then the relative uncertainty in x is   and the relative uncertainty in y is . The relative uncertainty in x·y (or x/y) is . Finally we can find the absolute uncertainty for s,

·        The relative error in  is relative error in x divided by 2.

 

 

Lab Report Preparation

     Write an appropriate lab report. It has to contain

1.    Your name and name of all your partners, date and time of work.

2.    Name of the lab (title).

3.    Goal of the lab.

4.    List of equipment used in the lab.

5.    Obtained data with detailed analysis.