1. CHAPTER 3 Units

2. 3.1 Units: What are they good for? Well, that is a very good question indeed. Let me begin by asking another question. If I were to tell you that a dog is 12 long, would you really know how long that dog was? Now, if I were to tell you that the dog was 12 inches long, you would have a much better idea of the size of the dog. That, in essence, is what units give us.

3. They give us a common standard to which we can compare other things. Since we all know how long one inch is, we can use that knowledge to see how long a 12 inch dog is. Most of us know English units fairly well, but they are not the most ideal units to work in because there are some weird units such as bushels that just don’t make any sense to most folks.

4. An internationally agreed upon standard system of units is needed to measure and represent a wide variety of physical quantities. Scientists have accepted the SI System for this purpose. The SI (Systeme Internationale) is founded on seven SI base units for seven base quantities assumed to be mutually independent, as given in Table 1. These are fundamental units of measurement.

5. Length Length is measured in a unit called the metre (often shortened to m). A door knob is usually about 1 metre from the ground: doorways are about 2 m high. For shorter distances we often use centimetres (100 cm = 1 metre) or millimetres (1000 mm = 1 metre). Question 1: Look at a metre rule. Which marks are centimetres and which are millimetres?

6. Length When measuring in Physics we try to do it as accurately as we can. Professor Messer is trying to measure the length of a block of wood with a metre rule but he has made at least six mistakes. Question 2: How many mistakes can you find?

8. Mass If you buy a bag of sugar in a shop, you will find the mass of sugar marked on the bag. It is written in grams (g) or in kilograms (kg). ‘Kilo’ always means a thousand, so 1 kilogram = 1000 grams. The mass of this laptop is about 3 kg. People often get confused between mass and weight, but they are not the same. Why?

9. Some important prefixes

10. Time In Physics, time is always measured in seconds (sometimes shortened to s). You can count seconds very roughly, without a watch, by saying at a steady rate: ONE (thousand) TWO (thousand) THREE (thousand) FOUR…

11. Experiment 1 Use a stop clock or stopwatch to measure the time for a complete swing of a pendulum or the beating of your heart. What is the time for 100 of your heartbeats? What is the time for one heartbeat? By how much does it change if you run upstairs?

12. All the other units you will meet this year are based on the metre, the kilogram and the second. They are called SI units. You might be surprised to know that, with these units, we can derive the majority of the other units that we need.

13. For instance, if we want the units for area, it is just “m  m”. This should be obvious because for example, the area of a rectangle is given by the formula, length  width. Since length and width would just be measured in metres, the result is that area has units of “m  m”.

14. Likewise, since the volume of a box can be determined by the formula, length  width  height, it should not come as a surprise that the units for volume is “m  m  m”. As you have probably guessed by now, we can just derive the units of a new quantity by looking at the formula or how it is defined. Let us do that for several of the things that we have already talked about. Namely, let us do this for speed, acceleration, and force.

15. 3.2 Very large and small numbers For very large or very small numbers, we sometimes use a shorthand way of writing them, by counting the number of zeros. For example: a)1 million = 1 000 000 (6 zeros) = 10 6 b)0.000 001 = (1 millionth) = 10 -6

16. In this shorthand way, write down: a)one thousand, b)one thousandth, c)10 million, d)one hundredth.

17. 3.3 Time of important events

18. SI base units You may find that some resources also refer to fundamental units as “basic” units or “base” units.

19. SI derived units Other quantities, called derived quantities, are defined in terms of the seven base quantities via a system of quantity equations. The SI derived units for these derived quantities are obtained from these equations and the seven SI base units. Examples of such SI derived units are given in the table below

23. Example 1 Prove that the derived unit of force is kg m s -2 (kg m/s 2 ) The equation of force is;Force = Mass  Acceleration Where Acceleration = Distance  Time Squared SoForce = Mass  Distance  Time Squared Units of Mass = kg Distance = m Time squared = s 2 SoForce = kg  m  s 2 = kg m s -2

24. Exercise 1 1)Prove that the derived unit of speed (or velocity) is m s -1 (m/s). Recall that speed = distance over time. 2)Prove that the derived unit of acceleration is m s -2 (m/s 2 ). Recall that acceleration = velocity over time.

25. 3.4 Important Notes about Units The order of magnitude of a number is the value of the number when rounded to the nearest power of ten. The order of magnitude of a number in scientific notation should be rounded up if the mantissa is larger than 3.16. (10 0.5 = 3.16) Prefixes are used in the SI system to serve as multipliers of fundamental and derived units.

26. All measurements include a value and a unit. Some also require a direction. Scientific notation enables very large and very small numbers to be expressed conveniently. Every measured quantity contains uncertainty. Uncertainty is usually expressed either in absolute terms or as a percentage.

27. Measured quantities should be expressed to the number of significant figures which best represent the accuracy of the measurement. When arithmetic operations are performed on measured quantities, the resulting answers should be expressed to an appropriate number of significant figures. In many experiments, measurements are taken of two or more variables to search for patterns or relationships which govern the way things behave.

28. Various techniques are used in physics to gather and interpret data. Data which are arranged in tables and then plotted on graphs can assist in the interpretation of those data. Computers are useful tools in manipulating and analysing data.

29. The shape of a curve on a graph may help to suggest a relationship between variables. Reading data from graphs is essential in interpreting numeric information. Interpolation and extrapolation are useful in interpreting graphical information.

30. 3.5 Converting Units Now, if everything in the world was easy, then we would only have to deal with one set of units. Alas, such is not the state of the world, and on occasion, we will have to convert units from one system to another. One example of such a conversion is finding out the metric equivalent of 12 feet. In order to do this, we need to first know the correct unit conversion factor. Well, it turns out that 1 feet is equal to 0.3048 metres.

31. Conversion factor: 1 feet = 0.3048 metres Converting units is not a hard thing to do. In fact, it really just involves multiplying and dividing. The best way to learn is to do an example, so here it goes.

32. Example 2: How many metres does 12 feet equal? Answer: The method is simple. Just change the word feet into its equivalent form i.e. 1 feet = 0.3048 metres So12 feet = 12  0.3048 metres = 3.6576 metres

33. Example 3: How many feet does 3 metres equal? We know that 1 feet = 0.3048 metres So the other way round becomes So3 metres = feet = 9.8425 feet

34. Example 4: What is 55 mph in terms of metres/sec? We need to know the conversion factor from miles/hour to metres/second 1 mile = 1609.344 metres and 1 hour = 60  60 = 3600 seconds. So

35. As stated above, we needed two conversion factors, which we multiplied to the 55 miles/hour. The first conversion factor converted the “miles” on top to “metres”. The second conversion factor changed the “hour” on the bottom to “seconds”.

36. Example 5: Convert 10 metres/sec to miles/hour We know that 1 mile = 1609.344 metres and 1 hour = 60  60 = 3600 seconds. So and

37. So

38. Exercise 2 Follow up the above examples to try to work out the following. 1)Convert 7.5 metres to feet 2)Convert 55miles to metres 3)Convert 75 miles/hour to metres/second 4)Convert 25 metres/second to miles/hour