Foundation Package THE END OF EDUCATION IS CHARACTER Foundation Package 1
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Laws of Indices Foundation Package 4
Logarithms Consider the following a x = b x is called the power or index. a is called the base. b is called the value or number. So, by definition: The log (short for logarithm) of a number is the power to which the base must be raised to equal the given number. Foundation Package 5
So log a b = x Read as the log of b to the base a equals x. e.g. If 10 2 = 100 Then log 10 100 = 2. If log 10 100 = 2, then the anti-log (2 nd function log in calculator) of 2 is 100. Any number can be chosen as the base, but base 10 and base e are commonly used. When the base is 10, it is written as log (or lg) where the 10 is understood. When the base is e (called natural logs), it is written as ln, where base e is understood. The number e is a special number. Foundation Package 6
Laws of Logs Foundation Package 7
Equation of a Straight Line The general equation of a straight line is y = mx + c. Where m is the slope of the graph, and c is the value of y when x = 0 Note: anything. 1. m, the slope, is the coefficient of the variable plotted on the x-axis, which could be 2. c is NOT the intercept on the y-axis. This is so only if the x-axis starts at zero. If the x-axis does not start at zero, then c has to be worked out, from the equation. Foundation Package 8
Note the Following Special Lines. 1. y = mx This is a straight line passing through the origin and going through first and third quadrants. 2. y = mx This is a line passing through the origin, going through the second and fourth quadrants. Foundation Package 9
3. y = x has a positive slope of 1. (bisecting first and third quadrants) 4. y = x Foundation Package 10
5. y = k A horizontal line, parallel to the x-axis i.e. slope is zero. 6. x = k A line parallel to the y-axis i.e. infinite slope. 7. y = 0 This is the x-axis. 8. x = 0 This is the y-axis. Foundation Package 11
Guidelines for Practicals Display of results (1) Use of table: Columns with appropriate headings, and units (2) Use of graph: Foundation Package 12
Plotting Graphs Students must be familiar with equations of the form: (a) y = ax n Using logs to obtain a straight line, a graph of log y vs log x is drawn. log y = log a + log x n log y = n log x + log a. Where n is the slope of the line, and log a is the intercept. Foundation Package 13
Note: when columns and axes are to be labelled with log or ln, they should be written as LOG (T/S) (1) It should be noted that unless otherwise stated, quantities should be given and used in S.I. units. (2) When logs are taken it is possible for all numbers to be negative. The graph may be entirely in the third quadrant. (3) You may not need to start a graph at the origin even if intercept is asked to be determined. Use an appropriate scale to make the graph as big as possible. The intercept is found by finding slope and substituting a point on the line in the equation of the line. DO NOT USE BROKEN SCALES Foundation Package 14
(b) Equations of the form: y = ae kx This time take natural logs i.e. ln. ln y = kx + ln a Students must be able to manipulate any type of equation to give a straight line if necessary and must be able to recognise the shape of a graph from the equation, if given. Foundation Package 15
Plotting Points It is recommended that points be plotted with small x s, and must be accurate within 2.0 mm on the graph page. So even if there are decimal places, one must still be able to determine within one small box. Drawing the Best Fit It should be noted that a graph is not just a connection of points, but a smooth curve is drawn through as many points as possible. If points do not fall exactly on the curve, then as far as possible, the sum of the deviations on either side of the curve is about the same. However, if a point is way off then you should recheck it and if no adjustment is made then neglect it. A smooth curve is drawn with a fine point pencil by drawing once without lifting the hand. If the graph is a straight line then a ruler must be used. If not, the graph is drawn free hand or with whatever curve-drawing instrument is available. Foundation Package 16
Calculations From the Graph (a) Slope: You can be asked to calculate the slope of the line or if it is a curve, the slope at a point. If you have to calculate the slope of the line then more than three quarters of the line must ( form the hypotenuse of an indicated 3 4) right-angled triangle. The points chosen must be points on the line and NOT necessarily points on the table. The units of the slope must be obtained from the units of the axes and the number of significant figures must be consistent with the number of significant figures in the observed results. Note: If it is a curve, and the slope has to be determined at a point, a tangent is drawn and the slope of the tangent is found. (b) Intercept: If the origin is not included in the graph then the intercept has to be calculated using the equation of the line. After the slope is found, choose a point on the line and substitute in the equation of the line to find intercept. Foundation Package 17
Scaling the Graph Page (i) Use the thick lines on the graph page as boundary lines for the axes. (ii) Use values for each centimetre or two centimetre blocks, such that it is easy to interpolate and determine what the value 2 mm on each axis represents. Foundation Package 18
Graph Analysis 1. Take note of the names and units of the axes. 2. Determine and interpret intercepts on the axes. 3. Determine and interpret the slope of the graph. The units will help. 4. Determine and interpret the area under a graph. The units will help. 5. Determine turning points maximum and minimum. 6. Determine and interpret any asymptotes. Foundation Package 19
The Nine Point Code of Conduct for Planning and Design Questions Physics Practicals 1. Identify the task this includes identifying or formulating the hypothesis to be tested. This becomes the aim of the experiment. 2. Identify the variables that are involved and identify which ones would have to be constant and which ones will vary. The aim will give a guide to this. Identify appropriate instruments to measure the variables that will be involved in the analysis. 3. Identify the independent and the dependent variables. Foundation Package 20
4. Identify appropriate operating equations and method of analysis that will be used to achieve the aim. 5. Brainstorm the difficulties that can arise if the experiment is to be done and find ways to overcome them. 6. Draw a diagram (or diagrams) of the set up. 7. For procedure: Write in sequence what is to be done. Identify what is to be measured and how. What difficulties are likely to be experienced and what steps can be taken to overcome them. 8. Calculations: Say how your results will be used to achieve the aim use of formula, graph and its analysis, etc. 9. Analysis/interpretation and conclusion. Foundation Package 21
Plotting of Linear Graphs You should be able to: rearrange relationships between physical quantities so that linear graphs may be plotted. Suppose we want to study the relationships between the time taken, t, for a ball that is thrown upwards to reach the ground and the ball s initial height, h, above the ground. We form a hypothesis that t is related to h by the equation 1 t = kh, where k is a constant of unknown value. How do 2 we test the validity of this equation? First, a few sets of readings of h and their corresponding t values have to be obtained. Then a graph of t versus h is plotted. If the above equation is true, we should obtain a curve. However, since many other equations will also give curves when plotted, we cannot conclude based on the curve alone, that the equation is valid. Figure 1 Graphs of t vs. h We cannot conclude from 1 this graph that t = kh, 2 as other equations might also give a similar curve. Foundation Package 22
To confirm the validity of the equation, we need to plot t versus h. If 1 the graph of t versus h gives a straight-line graph that passes through the origin, we know that the equation is valid. This is because the equation is now in the form of Y = mx (which is the equation of a straight line passing through the origin), where y = t, x = h, and m (the gradient of the graph) = k. So if 1 2 2 the plotted graph gives a straight line passing through the origin, we know that the equation is valid. 1 2 Figure 2 Graphs of t vs. h With this graph, we can conclude that the equation t = kh, is valid. 1 2 1 2 Foundation Package 23
In general, we can only conclude that a certain equation is valid, after we convert it into a straight-line equation, and upon plotting it, obtain a straight-line graph. The gradient and vertical intercept of the graph might yield useful information; in the above example, we are able to get the value of k from the gradient. Convert the following equations so as to get straight line graphs. Foundation Package 24
SOLUTION Foundation Package 25
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Measurement Techniques Errors and Uncertainties In an experimental work there will always be some uncertainty in measurements that we take. There are two categories of errors that we can talk about (1) random errors and (2) systematic errors. In scientific terminology, measurements and readings have different meanings. Reading: Is a single determination of the value of an unknown quantity. It s the actual reading taken during an experiment. Foundation Package 27
Measurement: Is the final result of the analysis of a series of reading. A measurement is only accurate up to a certain degree depending on the instrument used and the physical constraint of the observer. Any quantity measured has an amount of uncertainty or error in the value obtained. Note: If a rod is measured and its length is 34.7 cm, it indicates that it is only accurate to 0.1 cm, therefore to indicate the uncertainty of this value, it may be written as 34.7 ± 0.1 cm. The value of 0.1 cm is the absolute error in the measurement of the length of 34.7 cm. Error can also be stated in the form of fractional error. The fractional error in measurement of (34.7 ± 0.1)cm is whereas the error 0.1 34.7 Foundation Package 28
Systematic Errors Systematic errors are uncertainties in the measurement of physical quantities due to instrument faults in the surrounding conditions. One important characteristic of systematic errors is that the size of the error is roughly constant and the measurement obtained is always greater or less than the actual value. Foundation Package 29
Sources of Systematic Errors 1. Zero Errors: Occurs if the reading on an instrument is not zero even when it is not used to make any measurements e.g. hand of stopwatch does not point to zero but 0.2 seconds. 2. Personal error: results from physical constraints or limitation of an individual e.g. the reaction time. 3. Errors due to Instruments: (i) A fast watch (ii) an ammeter which is used under different conditions from which it had been calibrated e.g. Ammeter made in Japan has been calibrated under diff. Temp and earth s magnetic fields from Singapore where it is used. 4. Errors due to wrong assumption: Assuming g = 10 ms 2 whereas in reality it is 9.81 ms 2. Systematic errors cannot be reduced by taking a large number of readings using the same method, same instrument and by the same observer. Foundation Package 30
Random errors Random errors are uncertainties in a measurement made by the observer or person who takes the measurements. The characteristic of random errors is that it can be positive or negative and its magnitude is not constant. Thus the reading can be sometimes greater than the actual value and some times smaller than the actual value. Foundation Package 31
Examples of random errors: 1. Errors due to parallax when reading a scale. 2. Changes in temperature during an experiment can result in measurements being sometimes bigger and sometimes smaller than the actual value. Accuracy means that the mean value of the reading taken is close to the correct value even if the spread is wide. Precision means that there is little spread about the mean position of the values even though the mean of the values. Foundation Package 32
Error Calculations Percentage error in a calculated value is the sum of the % errors in the measured values. Foundation Package 33
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Graphs showing Accuracy and Precision Foundation Package 36
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How to reduce random errors: 1. In the case of low battery, replace the battery. 2. Repeat the experiment many times. How to reduce systematic errors: 1. Recalibrate instruments frequently. Foundation Package 38
Units and Dimensions Most physical quantities have units. Systeme Internationale (SI) Units The International System Of Units (French: Le Systemme International d Unites) was established by international agreement, and is widely used in many countries. The Caribbean has adopted this system. Foundation Package 39
S.I. Base Units Foundation Package 40
Derived units come from the base units. e.g. Change Pascals to base units. Foundation Package 41
Equations must be dimensionally consistent i.e. the net units on the right hand side must be equal to the net units on the left hand side. Because equations must be homogeneous, units can be used to check equations and even derive equations. Only like quantities can be added to each other or subtracted from each other. Foundation Package 42
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The Mole and the Avogadro Constant The Avogadro constant is a number that is often used by chemists. It is defined as the number of atoms in 12 g of carbon-12. This number is 6.02 10 23 (to 3 significant figures). The mole is the SI unit for the amount of substance. One mole is defined to be the quantity of substance containing a number of particles equal to the Avogadro constant. The idea behind the mole is similar to that of a dozen. One dozen of apples is 12 apples. One mole of atoms is 6.02 10 23 atoms. The mass of one mole of atoms or molecules of a substance is called the molar mass of the substance. Foundation Package 46
Example One mole of oxygen molecules has a mass of 32 g. Find: 1. the number of moles in 1.0 kg of oxygen, 2. the number of molecules in 1.0 kg of oxygen. Solution 1. Number of moles = 31 2. Number of molecules = 31 6.02 10 23 = 1.9 10 25 Foundation Package 47
Prefixes Scientists deal with quantities that are very big, for example the mass of the Earth, and quantities that are very small, for example the size of an atom. In order to facilitate recording these values as well as reading them, scientists use certain prefixes. Prefixes are used to denote multiplication by factors of 10. The prefixes with the corresponding factors of 10 are shown below: Foundation Package 48
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Questions on Units and Dimensions Foundation Package 50
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