01 Foundation package - Mr. Gopie Class · Foundation Package 8 Equation of a Straight Line The general equation of a straight line is y = mx + c. Where m is the slope of the graph,

Foundation Package 1

Foundation Package

THE END OF EDUCATION IS CHARACTER




Laws of Indices


Logarithms

Consider the following

ax = 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.


So loga b = x

Read as “the log of b to the base a equals x.” e.g. If 102 = 100 Then log10 100 = 2.

If log10 100 = 2, then the anti-log (2nd 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 under-stood. 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.


Laws of Logs


Equation of a Straight LineThe 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: 1. m, the slope, is the coefficient of the variable plotted on the x-axis, which could be anything.

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.


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.


3. y = x has a positive slope of 1.

(bisecting first and third

quadrants)

4. y = –x


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.


Guidelines for PracticalsDisplay of results

(1) Use of table: Columns with appropriate headings, and units

(2) Use of graph:


Plotting Graphs

Students must be familiar with equations of the form:

(a) y = axn

Using logs to obtain a straight line, a graph of log y vs log x is drawn.

log y = log a + log xn

log y = n log x + log a.

Where n is the slope of the line, and log a is the intercept.


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”


(b) Equations of the form:

y = aekx

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.


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.


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 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.

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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.


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.


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.


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.


Plotting of Linear GraphsYou 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 t = kh , where k is a constant of unknown value. How do 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 this graph that t = kh , as other equations might also give a similar curve.

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To confirm the validity of the equation, we need to

plot t versus h . If 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

the plotted graph gives a straight line passing through

the origin, we know that the equation is valid.

Figure 2 Graphs of t vs. h

With this graph, we can

conclude that the equation t

= kh , is valid.

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12

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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.


SOLUTION



Measurement TechniquesErrors 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.


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


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.


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.


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.


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.


Error Calculations

Percentage error in a calculated value is the sum of the % errors in

the measured values.




Graphs showing Accuracy and Precision



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.


Units and DimensionsMost 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.


S.I. Base Units


Derived units come from the base units.

e.g. Change Pascals to base units.


❑ 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.





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

× 1023 (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 × 1023 atoms. The mass of

one mole of atoms or molecules of a substance is called the molar mass of

the substance.


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 × 1023

= 1.9 × 1025


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:



Questions on Units and Dimensions


Documents

01 Foundation package - Mr. Gopie Class · Foundation Package 8 Equation of a Straight Line The general equation of a straight line is y = mx + c. Where m is the slope of the graph,