Revision:Measurement

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1.1 Standards of measurement

1.1.1 : Fundamental and derived units

Fundamental units are, in general those which can't be expressed in terms of others (there are exceptions). Mass, length, time, electric current (this is defined in terms of force between wires, but is fundamental in terms of electric circuits). The newton is a derived unit, because it is defined as the force required to accelerate 1kg at 1 ms^-2. Other derived units include Power (work x time), Pressure (force per unit area), density (mass per unit volume).

1.1.2 : Definition of some fundamental units

Kilogram...a measure of mass, defined by a platinum-iridium cylinder kept in Sevres, France (Though I really can't imagine the IB asking that :)

Meter...unit of distance, defined as the distance traveled by light in sec, where the speed of light (about 3 x 10⁸ m/sec).

Second...unit of time, based on time taken the number of vibrations in a cesium atom (9.1 x 10).

1.2 Vectors and scalars

1.2.1

Vector quantities have both a magnitude, and a direction. Scalar quantities have only a magnitude. Vector quantities are those such as displacement, velocity, acceleration. Scalar quantities are distance, speed, work and energy (those last two are important...apparently)

1.2.2

Vectors can be represented as lines, where the length is the magnitude and the direction is the direction on the paper. Vectors can be added by using a scale diagram...The first vector is drawn, then the second from the end of the first, and so on. The resultant vector goes from the beginning of the first to the end of the last...in that direction, not the other way.

1.2.3

Multiplying or dividing a vector by a scalar only affects the magnitude, not the direction...and works just like normal multiplication / dividing.

1.2.4

Place the vector as a diagonal of a rectangle...this allows the vertical and horizontal components to be calculated by pythag and basic trig. The sum of all the vertical components = the vertical component of the resultant vector and so for the horizontal.

1.3 Graphical techniques

1.3.1

Graphs should be drawn with the dependent variable on the vertical axis (unless the slop is supposed to be a particular unit over another, in which case use that). Usually only the dependent variable uncertainties are relevant, which means you only need vertical uncertainty bars ( and make sure you have a title, and label both axes...it's not like it requires any skill, and yet they still give marks for it :) Draw a line of best fit, usually a straight line, but not always...some points will probably have to be discarded, just to make things fun.

1.3.2

The units of the constant defining the slope of the graph will be . The range of possible slopes can be found by taking a maximum line of 'best' fit and a minimum line of 'best' fit using the uncertainty bars...Physics doesn't have to obey the foolish laws of grammar :)

The intercepts' relevance varies from graph to graph...in general, the intercept is the value of one component when the other is zero...ie on a temp vs pressure graph, the intercept will be a -273 c...representing absolute zero.

1.3.3

By playing around with powers (including negative powers) you can get a linear graph, from which it is much simple to determine the relationship. When you have a straight line which goes through the origin, the unit on the vertical axis is directly proportional to that on the horizontal axis.

1.3.4

Any straight line graph can be put in the form , where the slope and the intercept. Nb...If is not zero, then they are not directly proportional.

1.3.5

sin, or other repeating graphs have the following characteristics...amplitude -- the difference between the highest and lowest y values...Wavelength is the distance from the top of the crest of a wave to the top of the next crest (or equivalently, the distance between successive identical parts of a wave.) Period is the time required to complete one cycle ex. time for a pendulum to make one back and forth swing... frequency -- usually relevant in graphs against time, where frequency is the number of cycles per second...

1.3.6

Draw bar graphs...choose the appropriate intervals (they should all be of the same width, not too large or small to mask trends) and then find trends...Millikan's oil drop...the bars all differ by the same amount (the charge of an electron)...the frequency of values may increase or decrease with larger, or smaller values.

1.4 Uncertainties and errors

1.4.1

Uncertainties are due to lack of precision in measuring equipment, errors are actual inaccuracies ie equipment being mis-used, or mis-measurements...Uncertainties could come from the fact that a rule is only marked down to 1 mm, Errors could come if you miss read 15 on the ruler as 14...Uncertainties cause uncertainty bard, errors usually result in the particular piece of data being discarded.

1.4.2

Random uncertainties result from the magical randomness of measuring equipment...sometimes the jaws of a micrometer will close one way, sometimes another...they're random, and you can't do anything about them. Systematic errors are those built into the equipment.

1.4.3

Record uncertainty along with data. The minimum uncertainty is half the limit of the reading...ie if the measurement is 3.6g, then the uncertainty is ± 0.05g.

1.4.4

Random uncertainties are found by measuring the greatest difference from the arithmetic mean of the values...this decreases, at first rapidly and then more slowly as more data is collected. By using graphs we can obtain a line of best fit which wits within all the uncertainties.

1.4.5

When adding or subtracting, the uncertainty is the sum of the absolute uncertainties for each term. When multiplying or dividing, the uncertainty is the sum of the relative uncertainties (ie )...this can result in large uncertainties being created by performing operations on data with small uncertainties.

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