Introduction

Summary and Practice Applications of Fundamental Skills

general remarks:

Physics is the science, which addresses the interaction between matter and energy.  Physics is the science, which attempts to explain all of the fundamental truths of the observable universe.  The preferred language of physics is mathematics.  A physicist attempts to explain and as much as possible predict physical phenomena by incorporating quantitative information into mathematical relationships called equations.  So you should develop your quantitative skills to the fullest extent of your abilities.

Arguably, all of science can be considered branches of physics, however we will be kind to biologists, chemists, meteorologists, and others and let them think that they are pursuing separate academic disciplines.

The ultimate goal of physics is to find and develop an explanation of the entire universe and hopefully express it as an equation that will fit on a T-shirt.

measurement:

Measurement is the process by which people express quantitatively to each other aspects of the observable universe.  In order to engage in the process of measurement a common standard of measure must be used.  Ancient measurement systems used objects or parts of the human body for standards.

The metric system (i.e., System International) is used by the scientific community for the measurement of physical quantity.

The metric system uses seven fundamental units (see the chart below) to measure all aspects of the physical world.

All of the fundamental units are conceptual units (units based on repeatable physical phenomena such as the wavelength of light etc.) except mass.  Mass is still based on a standard kilogram mass kept in the International Bureau of Weights and Measures in France.

 Physical Quantity Symbol Used In Equations Unit Of Measure Unit Symbol length L meter m mass M kilogram kg time T second s electric charge Q coulomb Coul temperature T degree Kelvin K amount of substance (none) mole mol luminous intensity I candela cd

Fundamental units can be combined to form derived units.  For example: speed is the rate at which a moving body covers ground.  The derived unit used for speed therefore is the meter per second (m/s).  This unit is a combination of the fundamental units for length of distance (meters) and for elapsed time (seconds).

Units in the metric system are modified by prefixes which indicate the order of magnitude of the base unit.  An example is centi which stands for 1/100 of the base unit.  One centimeter represents 1/100 of a meter.

Significant Digits:

Significant digits in an experimental measurement are all the numbers that can be read directly from the instrument scale (known with certainty) plus one estimated (doubtful) number.

 Rule Example All non-zero digits are significant. 1.345 Kg has 4 significant digits All zeros between non-zero digits are significant 220005 Km has 6 significant digits Zeros to the right of a non-zero digit but to the left of an understood decimal are not significant. 230000 g has 2 significant digits Zeros to the right of the decimal but to the left of a non-zero digit are not significant 0.0000345 m has 3 significant digits Zeros to the right of the decimal following a non-zero digit are significant. 0.10070 cm has 5 significant digits

Multiplication And Division Using Significant Digits:

In the multiplication or division of two or more numerical measurements, the number of significant digits in the answer can be no greater than the least significant digits in any number in the set.
1. Determine the number of significant digits in each of the numbers in the problem.
2. Perform the multiplication or division.
3. Round the answer to the least number of significant digits determined by step one.

Addition And Subtraction Using Significant Digits:

In addition and subtraction, adding or subtraction begins with the first column from the left that contains an uncertain or doubtful figure.
1. Underline the first doubtful digit in each of the numbers in the set (this is the rightmost significant digit).
2. Add or subtract the numbers.
3. Round the sum or difference to the leftmost underlined place (doubtful figure).

Accuracy:

The closeness of a result to its accepted value is the accuracy of the result.  That which detrimentally affects accuracy is called error.  There are three sources of experimental error.

Human error sometimes called personal error arises from personal bias or carelessness in reading an instrument, in recording data, or in calculations.

Systematic error is associated with equipment problems such as an improperly zeroed instrument or an incorrectly calibrated device.

Random error results from unknown and unpredicted variations in experimental conditions.  Random errors are often beyond the control of the experimenter.  Voltage spikes, vibrations, and temperature fluctuations are all examples of the causes of random error.

Representation Of The Accuracy Of A Result:

Absolute error (Ea) is the numerical difference between the accepted value for a result and the actual experimental value.  Equation: Ea = Experimental value - Accepted value.

Relative error (Er) is a more meaningful representation of error in an experimental setting.  The relative error represents the fractional error.  You may have called this fractional error percent error.  Equation: Er = (Ea/Accepted) (100%).  (But both absolute and relative error presume you have a way to know the true or accepted value.  Often this is unavailable to scientists until measurements have been performed many times using a variety of procedures.)

Precision:

Precision is a measure of the reliability of a result.  It is measured by how close the measurement agrees with other measurements taken in the same way.  Poor precision scatters results and makes for an unreliable set of data.

Representation Of The Precision Of Data:

Mean or average can be computed for a set of data by using this equation: Xmean = ( X1 + X2 + X3 ... XN ) / N where X1 , X2 , etc. are the data points.  N is the number of data points in the data set.

Percent difference is a simple way to show agreement among data.
1. Find the mean of the data.
2. Find the difference between the two data points.  If there is a set of data find the difference between the biggest and smallest values (the extreme values in a set).
3. Divide the difference by the mean and multiply by 100%.
Equation for this: (X2 - X1 / mean) (100%) or (Xmax - Xmin / mean) (100%).

Absolute deviation represents the absolute value of the difference between a data point and the mean of the data.  Equation: δa = X - mean.

Mean absolute deviation represents the mean or average of the absolute deviations of a set of data.  When reporting the result of an experiment, the result may be expressed as: Mean of the data ± mean absolute deviation.  Example: 123 ± 2 cm.

Relative deviation represents the fractional deviation for a set of data.  To find the relative deviation use the following equation: ( mean absolute deviation / mean of the data ) (100%).  When reporting the result of an experiment, the result may be expressed as: Mean of the data ± relative deviation.  Example: 123 cm ± 0.5%.

Graphical Representation Of Data:

It is often convenient to represent experimental data in graphical form for the purposes of reporting and obtaining information (such as slopes).

Quantities are commonly plotted on Cartesian graph grids in which the horizontal (X) is called the abscissa and the vertical (Y) is called the ordinate.

Graphing Procedure:

1. Determine the range of data and choose scales that are easy to read and plot.  Scales which are too small will "bunch up" the data and make the graph unreadable.  Choose scales so that the major portion of the graph paper will be used.
2. Label each axis with the name of the quantity plotted (mass, time, velocity, etc.).
3. Indicate the units for the quantity in parenthesis (Kg), (s), (m/s), etc.
4. When plotting the individual data points, locate them as exactly as possible within the parameters of the scale.
5. When all of the data points are plotted, draw a smooth line connecting the points.  "Smooth" suggests that the line does not have to pass exactly through each point but connects the general areas of significance of the data points.
6. Title the graph.  The title is commonly listed as the Y quantity versus the X quantity.  Example: Distance vs. Time.
7. Put your name, date on the graph.

When two quantities are directly related the graph yields a straight line.  These quantities are considered to have a linear relationship.  The general equation is (Y = mX + b)

The slope of the line is useful in many physics applications.  The slope is found by dividing the Rise (Y) by the Run (X).  The unit used for the slope is determined by the X and Y units.  Example: X represents time (s) and Y represents distance (m), this would yield a slope which has the unit m/s.

If one quantity increases proportionally as the other decreases, an inverse relationship is present.  The graph of an inverses relationship is a hyperbola.  The general equation is (X)(Y) = k.

If one quantity increases as the square of the other then a quadratic relationship is present.  The resulting graph is a parabola.  The general equation is Y = kX2.

Problems In Data Analysis And Graphing

1. Find the absolute and relative error in a experiment which yields a result of 90.78 Units and has an accepted value of 88.00 Units.

2. An experiment is performed to find the density of a substance which has an accepted value of 1.23 g/cm3.  The mass of the test object is 239 g.  and the volume is measured as 190 cm3.  Compute the experimental density and the relative error of the result.

3. A wire with a cross sectional area of 3.55 X 10-5 m2 is broken by a force of 2340 N.  The accepted value for the tensile strength of the material is 6.60 X 107 N/m2.  Compute the experimental tensile strength and the relative error of the experiment.

4. An experiment is conducted twice yielding 23.44 Units and 24.05 Units.  Compute the mean and the percent difference for this set of data.

5. An experiment is performed eight times yielding results of 34.40, 34.80, 33.99, 34.00, 34.68, 34.05, 33.98, and 34.55 Units respectively.  Compute the mean and the percent difference for this set of data.  Use a table such as the one below:
 TRIAL DATA (UNITS) δa (UNITS) 1 32.56 2 32.55 3 31.98 4 32.01 5 32.48 6 33.00 7 32.85 8 32.56 mean

For the above data compute the mean, absolute deviation for each trial and the relative deviation for the experiment.  Complete the chart and report the experimental results using the mean absolute deviation and the relative deviation.
1. Graph the following sets of data on graph paper.
 MASS (Kg) WEIGHT (N) 2.2 21.5 2.5 24.5 2.8 27.6 3.0 29.4 3.1 30.4 3.2 31.3 3.4 33.3 3.6 35.2 3.8 37.2 4.0 39.2

2. Describe, in words, the relationship shown between mass and weight as shown on the graph.

3. Compute the slope of the line in this graph.
 VOL.  (cm3) MASS (g) 10.0 7.9 20.0 15.8 30.0 23.7 40.0 31.6 50.0 39.6

4. Describe the resulting curve.

5. Write an equation relating volume to mass.

6. Compute the slope of the graph.  What is the name given to this quantity?
 MASS (Kg) ACCEL.  (m/s2) 1.0 12.0 2.0 5.9 3.0 4.1 4.0 3.0 5.0 2.5 6.0 2.0

7. Describe the resulting curve.
8. Write an equation describing the relationship of mass to acceleration.
 FORCE (N) ACCEL.  (m/s2) 5.0 4.9 10.0 9.8 15.0 15.2 20.0 20.1 25.0 25.0 30.0 29.9

9. Describe, in words, the relationship shown between force and acceleration as shown on the graph.

10. Write an equation which shows the relationship.
 Volume Mass (kg) (cm3) Aluminum Water Wood 10 27 10 6 20 54 20 12 30 81 30 18 40 108 40 24 50 135 50 30 60 162 60 36 70 189 70 42 80 216 80 48 90 243 90 54 100 270 100 60

11. Plot the above data on the same graph.  Use different colors or symbols to indicate the materials.  Be sure to include a key on the graph to identify the three resulting lines.