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1.1 Standards of measurementEdit
Fundamental units are, in general, those which can't be expressed in terms of others (there are exceptions). The seven base units are: Mass(kg), length(m), time(s), electric current(amp) (this is defined in terms of force between wires, but is fundamental in terms of electric circuits), temperature(Kelvin), amount of substance (moles) and intensity of light (candela). 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 / time), Pressure (force per unit area), density (mass per unit volume).
Kilogram: A measure of mass, defined by a platinum-iridium cylinder kept in Sevres, France (Though I really can't imagine the IB exam asking that)
Meter: Unit of distance, defined as the distance traveled by light in 1/csec, where c = the speed of light (about 3 x 108 m/sec).
Second: Unit of time, based on the time taken for a cesium atom to vibrate about 9.1 x 109 times.
1.2 Vectors and scalarsEdit
Vector quantities have both a magnitude and a direction. Scalar quantities have only a magnitude. Vector quantities are those such as displacement, velocity and acceleration. Scalar quantities are mass, distance, speed, work and energy (those last two are important apparently).
"Scalar" magnitude only, can be described by single number and a unit e.g. speed (meter per second) temperature (K), time (sec), mass (kg), density (kg.m-3)
"Vector" magnitude and direction, can be described by two numbers or derived units e.g. velocity (meters per sec), force (N), acceleration (m.sec-2)
Vectors can be represented as lines, where the length is the magnitude and the direction is the direction on the paper.
Graphical representation of vector:
Length represents magnitude
Arrow represents direction
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).
Multiplying or dividing a vector by a scalar only affects the magnitude, not the direction. This works just like normal multiplication / division.
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Place the vector as a diagonal of a rectangle. This allows the vertical and horizontal components to be calculated via Pythagorus' theorem and basic trigonometry.
The sum of all the vertical components = the vertical component of the resultant vector and so for the horizontal.
1.3 Graphical techniquesEdit
Graphs should be drawn with the dependent variable on the vertical axis (unless the slope 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. Make sure to label both axes and title the graph. Draw a line of best fit, which is usually a straight line, but it is not always necessary. Some points may have to be discarded if they do not fit with the rest of the data.
The units of the constant defining the slope of the graph (also called the gradient) 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.
The intercepts' relevance varies from graph to graph. In general, the intercept is the value of one component when the other is zero, i.e. on a temp (x) vs pressure (y) graph for an ideal gas, the (x) intercept will be at -273.15 degrees Celsius, representing absolute zero.
By playing around with powers (including negative powers) you can get a linear graph, from which it is much simpler 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.
Any straight line graph can be put in the form y = mx + c , where m is the slope and c is the y intercept. Note: if c is not zero, then x and y are not directly proportional.
sin(x), or any other repeating functions have the following characteristics:
Amplitude: The difference between the highest, positive y value and the X-axis.
Wavelength: 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: The time required for one cycle. For example, the 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. frequency = velocity/wavelength.
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 errorsEdit
Uncertainties: are due to lack of precision in measuring equipment.
Errors: are actual inaccuracies i.e. equipment being mis-used, or mis-measurements.
Uncertainties can come from the fact that a ruler is only marked down to 1 mm. Errors can occur if you misread 15 on the ruler as 14. Uncertainties cause uncertainty bars, errors usually result in the particular piece of data being discarded.
There are different ways to measure uncertainties: with analog instruments, such as rulers, you would add onto the end of a value plus or minus half the value of the last digit, eg. on a ruler with 1mm precision, you would put +/- .5mm. Digital instruments use a different system, where it is plus or minus the value of the last digit, eg. with an electronic scale that reads 291g, the uncertainty would be +/- 1g.
Random uncertainties result from the 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.
Record uncertainty along with data. The minimum uncertainty is half the limit of the reading. i.e. if the measurement is 3.64g, then the uncertainty is ± 0.005g.
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 fits within all the uncertainties.
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 uncertainty/value). This can result in large uncertainties being created by performing operations on data with small uncertainties.
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