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SLIDE 1
1 mile = kilometres (approximately).
From this information we can create a conversion graph.
On the graph paper:
? Label the horizontal axis x / km.
Here x stands for the variable measuring a distance and km (kilometers) are the units of measurement.
? Label the vertical axis y / mi
mi is the standard abbreviation for miles
? Mark both axes in increments of 10. Plot the point and join this point to the origin .
SLIDE 2
? By drawing firstly a vertical and then a horizontal line onto the graph, use it to find to the nearest 0.5 mi how many miles is approximately equal to 100 km.
? By drawing firstly a horizontal and then a vertical line onto the graph, use it to find to the nearest 0.5 km how many kilometers is approximately equal to 42 miles.
SLIDE 3
?
?
SLIDE 4
Above is a currency conversion graph from US Dollars to Chinese Yuan.
? Use the graph to find what 85 US Dollars is worth in Chinese Yuan.
? Use the graph to find what 720 Yuan is worth in US Dollars.
? What is the exchange rate of $1 to Yuan?
? What is the exchange rate of 1 Yuan to $1? Give your answer to 3 significant figures.
SLIDE 5
?
?
? From the graph . The exchange rate is
? The exchange rate from Yuan to US Dollars is the reciprocal of the above,
SLIDE 6
A relationship represented by a straight line is called linear
If the relationship also passes through the origin it is directly linear proportional
Anything else involving a curve is non-linear
SLIDE 7
Classify each of the above as linear proportional, linear non-proportional or non-linear.
SLIDE 8
In the above we use the following shorthand: Proportional for linear proportional and linear for linear non-proportional.
SLIDE 9
In science (scientific explanation) the fundamental idea is that of cause (causation, causality).
We think that one thing causes another to happen.
The cause comes before the effect. Causes precede effects in time. Effects come after causes and are produced by them.
Not all relationships are causal relationships.
Examples
Causal relationship: a fall in temperature (cause) produces rain (effect)
Non-causal relationship: the exchange rate of the dollar to the pound is .
SLIDE 10
Classify each of the following as either a causal or non-causal relationship. If the relationship is causal, then state the cause and the effect.
? Kilometers, miles
? Volume of fuel in a car tank, distance travelled by the car
? Number of black counters in a box of counters, probability of picking a black counter.
? Smoking cigarettes, high blood-pressure
? Concentration of pollutant in a lake, proportion of dead fish
? Ticket sales in pounds, number of tickets sold
SLIDE 11
? Kilometers, miles ? non-causal relationship. It is a conversion.
? Distance travelled by the car causes the volume of fuel in a car tank to fall (effect) ? causal relationship.
? Number of black counters in a box of counters, probability of picking a black counter ? non-causal relationship. A probability is a proportion. It is not the effect of the number of black counters in the box.
? Smoking cigarettes causes high blood-pressure (effect) ? causal relationship.
? Concentration of pollutant in a lake causes the proportion of dead fish in the lake to increase (effect) ? causal relationship. A change in proportion can be the effect of a preceding cause.
? Ticket sales in pounds, number of tickets sold ? non-causal relationship. The two are correlated only. The increase in ticket sales in pounds does not precede the increase in the number of tickets sold as cause to effect. Advertising could cause both to increase.
SLIDE 12
Laws of nature
Whenever there is a causal relationship there is always a general pattern of like effects succeeding like causes.
Causal relationships are laws of nature.
Laws of nature involve the idea of something always happening in a regular way. In scientific explanation we do not have one-off causes.
One-off events may be caused. (They must be.) But the one-off event belongs to an underlying pattern of events.
Which of the following could be part of a statement a law of nature?
? The manager decided to install a new counter in the shop.
? Increase in temperature of air in a sealed can; increase in air pressure within the can; explosion of the can.
? The Duke of Wellington defeated Napoleon at the battle of Waterloo on 18 June 1815.
SLIDE 13
? The manager decided to install a new counter in the shop.
This is not a law of nature. It is a one-off event not describing a regular pattern.
? Increase in temperature of air in a sealed can; increase in air pressure within the can; explosion of the can.
Could be part of a law of nature. It is a law of nature that an increase in temperature brings about an increase in pressure. The explosion of the can is a one-off effect of this law.
? The Duke of Wellington defeated Napoleon at the battle of Waterloo on 18 June 1815.
Not a law of nature. Historical dates do not state causes.
SLIDE 14
Which of the following could be part of a statement a law of nature?
? Cystic fibrosis (a disease of the lungs), mutation of a gene regulating the production of a protein.
? Increase of voltage in an electrical circuit; increase in electrical current.
? If it was raining this morning, Mike has his umbrella. Mike does not have his umbrella. Therefore, it was not raining this morning.
SLIDE 15
? Cystic fibrosis (a disease of the lungs), mutation of a gene regulating the production of a protein.
This could be part of a law of nature. Mutations in genes are part of a regular pattern of explanation that accounts for genetic diseases.
? Increase of voltage in an electrical circuit; increase in electrical current.
This is an expression of a law of nature.
? If it was raining this morning, Mike has his umbrella. Mike does not have his umbrella. Therefore, it was not raining this morning.
This is not a law of nature. It is a logical deduction. That Mike always takes an umbrella, if it is raining could be part of a scientific explanation, but this is not stated in the information.
SLIDE 16
Could there be any event occurring that did not have a cause, and could not be explained by a law of nature?
SLIDE 17
Could there be any event occurring that did not have a cause, and could not be explained by a law of nature?
Miracles
There is no agreed answer to this question. Some people do believe that there are events, called miracles, that do not have any causes. Other people deny this.
The topic of miracles belongs to philosophy and theology.
Some people believe that science explains everything. Other people do not agree with this.
SLIDE 18
Each of the following belongs to a subject domain. State the subject, and state whether that subject is or is not a science.
? In the novel Pride and Prejudice by Jane Austen, the character of Darcey is proud, and Lizzie is prejudiced against him.
? An element with valence +1 forms a compound with an element of valence in the ratio 1 to 1. For example, sodium (symbol Na) has valence +1 and chlorine (symbol Cl) has valence . They form the compound sodium chloride with sodium to chlorine . The formula of sodium chloride is NaCl.
? In photosynthesis, plants make sugars from carbon dioxide and water using the energy of light trapped by the pigment chlorophyll.
SLIDE 19
? In the novel Pride and Prejudice by Jane Austen, the character of Darcey is proud, and Lizzie is prejudiced against him.
The subject is literary criticism, also called literature or critical analysis. People who write novels seem to think they can explain people’s behaviour in terms of folk psychology. However, this is not a science, and scientists do not recognize critical analysis or folk psychology as a science.
? An element with valence +1 forms a compound with an element of valence in the ratio 1 to 1. For example, sodium (symbol Na) has valence +1 and chlorine (symbol Cl) has valence . They form the compound sodium chloride with sodium : chlorine . The formula of sodium chloride is NaCl.
The subject is chemistry, and this is a part of science.
? In photosynthesis, plants make sugars from carbon dioxide and water using the energy of light trapped by the pigment chlorophyll.
The subject is biology, and this is a part of science.
SLIDE 20
Each of the following belongs to a subject domain. State the subject, and state whether that subject is or is not a science.
? Weight is the effect of gravity on small objects near the surface of the Earth. Gravity is an attraction between objects because of the amount of substance they contain, which is called mass.
? The Franco-Prussian war of 1870 to 1871 started after German Chancellor, Otto von Bismarck, edited the Ems telegram from King William I of Prussia to Emperor Napoleon III of France.
? Poverty is the cause of crime.
SLIDE 21
? Weight is the effect of gravity on small objects near the surface of the Earth. Gravity is an attraction between objects because of the amount of substance they contain, which is called mass.
The subject is physics, and this is a part of science.
? The Franco-Prussian war of 1870 to 1871 started after German Chancellor, Otto von Bismarck, edited the Ems telegram from King William I of Prussia to Emperor Napoleon III of France.
The subject is history. The statement involves explanation in terms of historical events and causes, but it is not a part of science. There is no general law expressed here, such as, every time Bismarck edits a telegram, there will be a war between Germany and France.
? Poverty is the cause of crime.
The subject is sociology. Not everyone agrees that poverty causes crime. Sociology is a soft science, meaning that it has laws of nature and uses scientific explanation, but not in the precise and mathematical way of physics, chemistry and biology, which are called hard sciences.
SLIDE 22
The hard sciences have laws of nature to which there are no exceptions that are often expressed in mathematical formulas.
The soft sciences study human nature. They use laws of nature, but these may have exceptions, and may not involve mathematical formulas.
The humanities are not sciences.
There also vestiges of ancient sciences, now no longer regarded as sciences, and called pseudo sciences by some philosophers.
SLIDE 23
Classify each of the following subjects as either a hard science, a soft science, a humanity or a pseudo-science.
? Physics ? Chemistry
? Biology ? Geology
? Psychology ? Sociology
? Economics ? Sport science
? Geography ? History
? Poetics ? Grammar
? Literary criticism ? Bible Studies
? Business Studies ? Philosophy
? Theology ? Astronomy
? Astrology ? Haruspicy*
* Haruspicy was the practice of divining the future by examination of such things as thunderstorms. It was practiced by the ancient Etruscans and Romans.
SLIDE 24
Hard sciences
Physics, Chemistry, Biology, Geology, Astronomy
Soft sciences
Psychology, Sociology, Economics, Sport science, Business Studies
Humanities
Geography, History, Poetics, Grammar, Literary criticism, Bible Studies, Philosophy, Theology
Pseudo-sciences or vestiges of ancient sciences
Astrology, Haruspicy
SLIDE 25
Laws of nature (scientific laws) are expressed as relationships between variables. For example
Law of Terrestrial Gravity
where in Newtons (N), in kilograms (kg) and , which is a constant.
The two variables are weight and mass.
Variables are quantities that can be measured. They have units of measurement. Here, weight is measured in Newtons (N), and mass in kilograms (kg).
With one exception* laws do not refer to time, and do not say that one variable is the cause of the other. But we tend to think of one as dependent (depending) on the other. In the example, we think of weight as caused by mass, and not the other way around.
Weight is the dependent variable, and mass is the independent variable.
* The Second Law of Thermodynamics, which is in advance of this chapter, is the only law in science (Physics) in which the direction of time is indicated.
SLIDE 25B
Laws of nature (scientific laws) are expressed as relationships between variables. For example
Law of Terrestrial Gravity
where in pound force (lbf), in slugs (slug) and , which is a constant.
The two variables are weight and mass.
Variables are quantities that can be measured. They have units of measurement. Here, weight is measured in pound force (lbf), and mass in slug (slug).
With one exception* laws do not refer to time, and do not say that one variable is the cause of the other. But we tend to think of one as dependent (depending) on the other. In the example, we think of weight as caused by mass, and not the other way around.
Weight is the dependent variable, and mass is the independent variable.
* The Second Law of Thermodynamics, which is in advance of this chapter, is the only law in science (Physics) in which the direction of time is indicated.
SLIDE 26
State in the following two examples of scientific laws, which is the dependent and which is the independent variable.
? Hooke’s Law for springs
where in metres (m), in Newtons (N), and .
? Ohm’s law for resistance
where in volts (V), in amps (A) and in Ohms , which is constant for a given resistor.
SLIDE 26B
State in the following two examples of scientific laws, which is the dependent and which is the independent variable.
? Hooke’s Law for springs
where in feet (ft), in pound force (lbs), and .
? Ohm’s law for resistance
where in volts (V), in amps (A) and in Ohms , which is constant for a given resistor.
SLIDE 27
? Hooke’s Law for springs
where in metres (m), in Newtons (N), and .
The dependent variable is x, extension, the independent variable is F, force.
? Ohm’s law for resistance
where in volts (V), in amps (A) and in Ohms , which is constant for a given resistor.
The dependent variable is I, current and the independent variable is V, potential difference.
Notes
These two laws are usually found in the form, and . Here we have rearranged them to make clear the relation between dependent and independent variable. The spring constant is the reciprocal of the constant k in . In the constant is called conductance.
SLIDE 27B
? Hooke’s Law for springs
where in feet (ft), in pound force (lbs), and .
The dependent variable is x, extension, the independent variable is F, force.
? Ohm’s law for resistance
where in volts (V), in amps (A) and in Ohms , which is constant for a given resistor.
The dependent variable is I, current and the independent variable is V, potential difference.
Notes
These two laws are usually found in the form, and . Here we have rearranged them to make clear the relation between dependent and independent variable. The spring constant is the reciprocal of the constant k in . In the constant is called conductance.
SLIDE 28
In a graph the independent variable is shown on the horizontal axis and the dependent variable on the vertical axis.
Law of Terrestrial Gravity
where in Newtons (N), in kilograms (kg) and , which is a constant.
Task
In the law of terrestrial gravity, the weight is directly proportional to the mass.
Draw a graph. Label the axes of the graph with the independent and dependent variable. Mark the axes from 0 to 100 Newtons and from 0 to 10 kg.
When the mass is 5 kg the weight is 49 N. Plot the point onto the graph and draw the line through this point and the origin. What is the weight of a mass of 8 kg to the nearest Newton?
SLIDE 29
Law of Terrestrial Gravity
SLIDE 28B
In a graph the independent variable is shown on the horizontal axis and the dependent variable on the vertical axis.
Law of Terrestrial Gravity
, where in pound force (lbf), in slugs (slug), , which is a constant.
In the law of terrestrial gravity, the weight is directly proportional to the mass. Draw a graph. Label the axes of the graph with the independent and dependent variable. Mark the axes from 0 to 200 lbf Newtons and from 0 to 5 slug. When the mass is 5 slug the weight is 161 lbf. Plot the point onto the graph and draw the line through this point and the origin. What is the weight of a mass of 3 slug to the nearest pound force?
SLIDE 29B
Law of Terrestrial Gravity
SLIDE 30
Law of Terrestrial Gravity
The constant g is called the gravitational field strength. The straight line, directly proportional, relationship indicates that g is always the same number.
When the mass is 5 kg the weight is 49 N. What is the value of the gravitational field constant?
SLIDE 31
The Law of Terrestrial Gravity is
SLIDE 32
The units of the gravitational field constant are written .
This is read “Newtons per kilogram”. In science the index is preferred to the division symbol. But in some texts, the division symbol is still used, and the units are written . The slash indicates division. is the reciprocal of kilograms.
has the same meaning as “Newtons per kilogram”.
Question
Express using a negative index
? 3 metres per second
? 1,500,000 Newtons per square metre
?
?
? acceleration
SLIDE 33
? 3 metres per second ?
? 1,500,000 Newtons per square metre ?
? ?
? ?
? acceleration ?
SLIDE 34
In a right-angled triangle the ratio of the rise to the step is called the gradient.
The gradient of the triangle on the right is
SLIDE 35
Find the gradient of each of the following triangles. Express the answers as fractions in their lowest terms.
SLIDE 36
? ?
? ?
?
SLIDE 37
where in Newtons (N), in kilograms (kg) and , which is a constant.
This is the law of gravity on the surface of a large mass, such as the Earth or Moon.
An experiment was conducted on the Moon with a rock. The mass of the rock was 1.5 kg. The weight of the rock was 2.55 N.
What is the gravitational field strength of the Moon?
SLIDE 38
An experiment was conducted on the Moon with a rock. The mass of the rock was 1.5 kg. The weight of the rock was 2.55 N. What is the gravitational field strength of the Moon?
SLIDE 39
The following table shows the weight of a 10 kg mass on various planetary bodies.
Earth 98 N
Moon 17 N
Mars 37 N
Venus 88 N
Plot the points , , and onto the graph. By joining each of these points by a straight line to the origin, find the gradients of each line, and hence find the gravitational field strengths of the Earth, Moon, Mars and Venus.
SLIDE 40
In each case the base of the triangle is 10. The heights are Earth 98, Venus 88, Mars 37 and Moon 17 respectively. The gravitational field strength of each is the gradient of these lines.
SLIDE 41
The equation of direct proportionality is
where k is the gradient of the straight line going through the origin.
We are often told that one variable is directly proportional to another. This statement is written
This uses the proportionality symbol .
This symbol looks a little like the infinity symbol, , but is open at the back end.
SLIDE 42
Example
The money received in ticket sales (S) at a theatre is directly proportional to the number to tickets sold (n).
When 1,200 tickets are sold, the sales are £11,820.
Find the constant of proportionality, and the equation connecting ticket sales to number of tickets.
Solution
STEP 1 S is proportional to n
STEP 2 Introduces the constant of proportionality
STEP 3 Substitute the values for S and n
STEP 4 Solve to find k
is the ticket price.
Equation:
SLIDE 43
Pounds sterling is proportional to Euros. When 200 euros are exchanged, £180 are received. Find the constant of proportionality and the equation connecting pounds (L) to euros (E).
SLIDE 44
Pounds sterling is proportional to Euros. When 200 euros are exchanged, £180 are received. Find the constant of proportionality and the equation connecting pounds (L) to euros (E).
STEP 1 L is proportional to E
STEP 2 Introduces constant of proportionality
STEP 3 Substitute the values for L and E
STEP 4 Solve to find k
STEP 5 Equation
SLIDE 45
At an American university, the grade result in the exam (R) of a student is directly proportional to the number of hours worked by that student in the final week before the exam (h), up to a maximum of 80 hours per week. One student, who worked 19 hours in the final week obtained a score of 23.75%. What is the equation connecting exam result R with hours worked in the final week, h? What would be the score of a student who worked 55 hours in the final week?
SLIDE 46
At an American university, the grade result in the exam (R) of a student is directly proportional to the number of hours worked by that student in the final week before the exam (h), up to a maximum of 80 hours per week. One student, who worked 19 hours in the final week obtained a score of 23.75%. What is the equation connecting exam result R with hours worked in the final week, h? What would be the score of a student who worked 55 hours in the final week?
Solution
When ,
SLIDE 47
We are given
Complete the following table and plot the graph of y against x.
x 0 2 4 6 8 10
y
What is the constant of proportionality and gradient of the line?
SLIDE 48
x 0 2 4 6 8 10
y 0 6.4 12.8 19.2 25.6 32.0
The constant of proportionality and gradient are both 3.2
SLIDE 49
? On a flat surface, how many points are needed to draw a straight line?
? How might the answer to the first question differ, if the line is a line of longitude drawn on the surface of a sphere?
? You are given that . In addition to the origin, how many pairs of values do you need in order to draw the graph of ?
SLIDE 50
? You only need two points to draw a straight line on a flat surface.
? Two points are not sufficient to determine a line of longitude if it is drawn on a sphere.
There are infinitely many lines joining the North pole to the South pole.
? In addition to the origin, you only need one pair of values to draw the graph of .
SLIDE 51
According to Hooke’s law the extension of a spring is proportional to the force stretching it.
In an experiment a spring was stretched, and the following graph was drawn.
Supposing that the spring obeys Hooke’s law, why aren’t all the data points lying exactly on the line?
SLIDE 52
The data points do not lie exactly on a straight line because of random error in the experimental procedure.
All experiments involve measurement, and measurement is only accurate to a degree. You must read a measurement with your eye, and that introduces differences that lead to variation. Also, the spring may have small defects in it that could cause it not to stretch proportionately. Hooke’s law is for an ideal spring – a spring that has no imperfections in it. Ideal springs do not really exist.
SLIDE 53
A spring can be stretched by attaching a hanger to it and adding masses.
In an experiment investigating how the length of the spring changes as more masses are added the following results were obtained.
Mass / g Length / cm Mass / g Length / cm
0 10.8 250 12.8
50 11.2 300 13.3
100 11.5 350 13.6
150 11.9 400 14.1
200 12.4 450 14.4
Find the extension of the spring for each mass, plot the graph of extension against mass including the line of best fit, find the gradient, the constant of proportionality, and the equation connecting extension (x) and mass for this spring (m).
SLIDE 54
Mass / g Length / cm Extension
/ cm Mass
/ g Length / cm Extension
/ cm
0 10.8 0 250 12.8 2.0
50 11.2 0.4 300 13.3 2.5
100 11.5 0.7 350 13.6 2.8
150 11.9 1.1 400 14.1 3.3
200 12.4 1.6 450 14.4 3.6
The equation is , where x is extension, m is mass
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