1. MEASUREMENT

1.1. What is physics?

universal:

Some sciences study specific objects or phenomena, for example

- The fish is an animal (biology)

- The stone consists of granite (geology)

- The battery is a source of electric voltage (engineering)

Physics (sometimes also parts of chemistry) studies properties which these have in common (universal phenomena)

- The stone/battery/fish weighs 50 g (physics)

- The stone/battery/fish falls down because of the force of gravity (physics)

- The stone/battery/fish consists of atoms (physics, chemistry)

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experimental:

This means that what is ultimately true is decided by experimental tests. The experiments are sometimes done by school students, but more often knowledge gained from experiments done by professional scientists is communicated to students. This can be compared to geography: you do not learn about South America in geography by sailing there and seeing it for yourself, you get and discuss a map based on observations made by others.

mathematical:

Both experiments and theories in physics often involve mathematical descriptions and analyses. The mathematics in IB physics is often not as hard as in the mathematics courses, but basic maths is used a lot.

1.2. The SI-system

Results of physical measurements are often reported as:

distance = 5.0 kilometers or d = 5.0 km

distance = quantity, d = symbol of quantity, 5.0 = value, kilometers = unit, km = abbreviation of unit, kilo = prefix, k = symbol of prefix

· Scientific notation : d = 5.0 × 10³ m, sometimes d = 5.0 x 10³ m or d = 5.0 * 10³ m

·        Prefixes, abbreviations and values:

tera = T = 10¹², giga = G = 10⁹, mega = M = 10⁶, kilo = k =10³, hecto = h = 10², deca = da = 10¹, deci = d = 10^-1, centi = c = 10^-2, milli = m = 10^-3, micro = m = 10^-6, nano = n = 10^-9, pico = p = 10^-12, femto = f = 10^-15

(DB p. 2)

· Fundamental SI-units:

Quantity Unit Symbol

Mass kilogram kg

Length meter m

Time second s

Electric current ampere A

Temperature kelvin K

Amount of substance mole mol

· Derived units : "meter per second" = ms^-1 (not m/s !) for speed = distance/time ; "kilogram per cubic meter" = kgm^-3 for density = mass/volume

1.3. Vectors and scalars

Types of physical quantities

Scalar quantity : has only magnitude, ex. time, mass, distance, temperature

Vector quantity : has both direction and magnitude, ex. force, velocity

If we only consider one dimension, then vector quantities can be mathematically treated as ordinary numbers where we let the sign indicate the direction. Ex. If a train moves at 20 ms^-1 forwards we denote the velocity v = +20 ms^-1 = 20 ms^-1, if it moves backwards v = - 20 ms^-1. In two (or, which is more rarely studied here, three) dimensions a vector can be symbolised by an arrow which indicates the direction and which has a length that shows the magnitude.

Note: Signs can be used also for scalar quantites, but then they mean something else than a direction: for example temperature can be described with "-10 ^oC" but the minus sign only means that the temperature is below the freezing point of water.

Graphical addition and subtraction of vectors:

Place the vectors A and B so that they start from the same point and form a parallellogram. The sum of them = A + B will be one diagonal, the difference A - B = A + (-B) will be another.

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Resolving a vector into components

Vectors can be resolved ("split") into components in two dimensions at a 90^o angle to each other. These two dimensions are chosen in a way that suits the problem under study (ex. horizontal and vertical; or north-south and east-west, or parallel to a slope and perpendicular to it.

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From the diagram above we find that:

sin q = opposite side / hypotenuse = A_V / A => A_V = A sin q_A

cos q = near side / hypotenuse = A_H / A => A_H = A cos q_A

tan q = opposite side / near side = A_V / A_H

A_H = A cos q_A A_V = A sin q_A DB p.4

Example: Let A = 5.00 m and q_A = 40^o. The sine (sin) and cosine (cos) values are obtained from an electronic calculator; sometimes by first punching in 40 and then sin or on others first sin and then 40. One must check that the calculator is set for the angle unit degrees, not radians or gradians.

We will then have A_V = 5.00 m * sin 40^o = 5.00m * 0.6427876... = 3.213938.. m = 3.214 m

and A_H = 5.00 m * cos 40^o = 5.00 m * 0.7660444... = 3.830222... m = 3.830 m

We can also check that tan 40^o = 0.83909963.... with a calculator or tan 40^o = A_V / A_H = 3.214 m / 3.830 m = 0.839164... (a small difference because of the approximation).

Adding the components of two vectors

If we have split the vectors A and B into the components A_H and A_V, B_H and B_V we can add the two vectors by adding the components. If we say that the vector C = A + B then C_H = A_H + B_H and C_V = A_V + B_V. (In the same way we could let D = A - B => D_H = A_H - B_H and D_V = A_V - B_V. The "negative" of a vector is a vector of the same magnitude = length but opposite direction).

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Let us say that we have B = 7.00 m and q_B = 65^o (In the graph above the angle between B and the horizontal axis seems to be much less than that, but when drawing a graph with partially unknown vectors it is not necessary for their lengths and angles to be accurate unless that is particularly requested). Then B_H = B cos q_B = 7.00 m * cos 65^o = 2.958 m and B_V = B sin q_B = 7.00 m * sin 65^o = 6.344 m. We will then get:

C_H = A_H + B_H = 3.830 m + 2.958 m = 6.788 m and

C_V = A_V + B_V = 3.214 m + 6.344 m = 9.558 m

Finding the magnitude and direction of the resultant

The sum of two or more vectors can be called a resultant. If we know the components C_H and and C_V of the resultant C then its length can be found using Pythagoras' rule:

C² = C_H² + C_V² => C = Ö (C_H² + C_V²) = (C_H² + C_V²)^½ = Ö( (6.788m)² + (9.558m)² ) = 11.723 m

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The angle q' can be found using the inverted functions of the sine, cosine or tangent. These are properly called arcsine (arcsin), arccosine (arccos) and arctangent (arctan) but on many calculators they are (not mathematically correctly) called sin^-1, cos^-1 and tan^-1. You can check that you have found the function on your calculator using what we found earlier:

sin 40^o = 0.6427876... so arcsin 0.642 = 39.9411^o

cos 40^o = 0.7660444... so arccos 0.766 = 40.00396^o

tan 40^o = 0.83909963.... so arctan 0.839 = 39.9966^o

Notice that sin 40^o is ca 0.642 so arcsin 0.642 = 40^o but (0.642)^-1 = 1/0.642 = ca 1.56, something not even near 40 !

Now for C we can use one of these:

sinq' = C_V/C => q' = arcsin(C_V/C) = arcsin(9.558m/11.723m) = arcsin(0.81532) = 54.62^o

cosq' = C_H/C => q' = arccos(C_H/C) = arccos(6.788m/11.723m) = arccos(0.57903) = 54.62^o

tanq' = C_V/C_H => q' = arctan(C_V/C_H) = arctan(9.558m/6.788m) = arctan(1.40807) = 54.62^o

1.4. Graphs

Linear graphs

In reports, refer to graphs as graph or diagram nr so-and-so. Remember to indicate units on the scales. The zero can be suppressed if only high values are used:

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· Slope or gradient : unit of vertical axis divided with unit of horizontal (here: the gradient is the accceleration a, and its unit is (ms^-1)/s = ms^-2 . Slope or gradient (m) and intercept (c) for straight line is given by y = mx + c (here we get v = at + u)

· Area under line or curve: units of axes multiplied (here the area under the graph is the displacement s; its unit is (ms^-1)*s = m

Transforming non-linear graphs to linear

If the graph is not linear from the start, then it can be made linear by plotting a manipulated variable on one or both of the axes. As an example, take s = ut + ½at² for UAM which becomes s = ½at² when u = 0. Assume that a = 2 ms^-2 so ½a = 1 ms^-2, then we can express the displacement as a function of time with data points, (time in sec, displacement in m ) = (t,s) : (0,0) , (1,1) , (2,4) , (3,9) etc. This graph is not linear, but if we instead plot (t²,s) we get (0,0) , (1,1) , (4,4) , (9,9) etc. which is a linear graph with the gradient 1 so ½a = 1 ms^-2 and a = 2 ms^-2.

Instead we could have plotted (t, Ös) giving (0,0) , (1,1) , (2,2) , (3,3) which also is linear and has the gradient 1 giving a = 2 ms^-2.

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In similar ways we can transform other mathematical features in a formula:

·        if we (for an ideal gas) have PV = nRT => P = nRT/V = k/V for constant n, R and T we get a straight line by plotting P as a function of 1/V

·        for logarithmic graphs, see the section about radioactive decay in nuclear physics later

·        Fitting a line to data points : one line, not pieces from point to point. As many points above the line as below.

Fitting a linear graph ("best-fit") to experimental data points

If we are working with experimental values that do or "should" follow a straight line (either as they are or after some mathematical manipulation like squaring them or taking the square root of them), then they may not exactly lie on a straight line, but we can fit a line to them by drawing one line that approximately follows them (possibly disregarding "outliers", individual values which are very different from the others and may be caused by mistakes in the experimental work), leaving about half of the data points below and above the line.

Note: Do NOT draw several lines that join all the data points !

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1.5. Errors and uncertainties

Types of errors or uncertainties

· Random uncertainty: always present in measurement, if you measure the lenght of an object many will say it is 24.7 cm but some 24.6 cm or 24.8 cm, some even 24.9 cm. Repeating measurements and taking the average will decrease the uncertainty first, but if you already have 5 measurements taking 5 more will not give a much better result. A low random uncertainty means we have a high precision.

· Systematic error: ex. if you measure the outside temperature with a thermometer in the sun. More measurements will not cure this. A low systematic error means we have a high accuracy.

distance = 5.0 ± 0.5 m or (5.0 ± 0.5)m

· absolute uncertainty : 0.5 m, that is the actual value can be between 4.5m and 5.5m

· fractional uncertainty : 0.5m/5.0m = 0.1

· relative or percentage uncertainty : 10%

Combining uncertainties in calculations:

If you add or subtract, add absolute uncertainty

If you multiply or divide, add fractional or percentage uncertainty

At the end, always give absolute uncertainty.

which can also be expressed as:

If y = a ± b then Dy = Da + Db

If y = ab/c then Dy/y = Da/a + Db/b + Dc/c DB p. 4

Example: The first part of the trip took 25±3 s, the second part 17±2s. How long time did the whole trip take? How much longer did the first part take compared to the second part?

Add values : 25s + 17s = 42s. Add absolute uncertainties: 3s +2s = 5s. Answer: 42±5s.

To answer the second question we need the difference which is 25s - 17s = 8 s. But we still add the uncertainties and get the answer 8±5s

Example: We covered 600±12 m in 30±3 s. What was the speed?

Speed = 600m/30s = 20ms^-1. Uncertainties: in distance 12m which gives 12m/600m = 0.02 = 2%, in time 3s which gives 3s/30s = 0.1 = 10%. Adding gives 12% uncertainty in speed. Now 12% of 20 is 2.4 so we get the answer 20±2.4ms^-1. In the final result the uncertainty is sometimes approximated to one significant digit, here this would give 20±2ms^-1.

Example: If we use the formula x = y/z² and the percentage uncertainty in y is 5% and in z 3%, what is it in x?

x = y/z² = y / zz so the percentage uncertainties are added: 5% + 3% + 3% = 5% + 2*3% = 11%

Example: Same as above, but the formula is x = y/Öz ?

x = y/Öz = y/z^½ so in analogy with the above, we get 5% + ½*3% = 6.5%

Example: If using the formula v = u + at we insert u = 5.0±0.5 ms^-1, a = 0.1±0.005 ms^-2 and t = 3±0.15 s, what will v be?

For the value we get v = u + at = 5.0 ms^-1 + 0.1ms^-2 * 3.0 s = 5.3 ms^-1

The relative uncertainty in a = 0.005/0.1 = 0.05 = 5% and in t = 0.15/3.0 = 0.05 = 5% so the relative uncertainty in at is 5% + 5% = 10%. Therefore the absolute uncertainty in at is 10% of the value of at = 0.1ms^-2 * 3.0 s = 0.3 ms^-1 which is 0.03 ms^-1 . When adding u and at we shall then add their absolute uncertainties so we get 0.5 ms^-1+ 0.03 ms^-1 = 0.53 ms^-1.

Therefore we finally get v = 5.3 ms^-1 + 0.53 ms^-1 which can be given as 5.3 ms^-1 + 0.5 ms^-1 .

Finding the absolute uncertainty in the measurement

In the examples above we have assumed that the absolute uncertainty in a value is given ("We covered 600±12 m in 30±3 s" - the 12 and the 3 are given). But how can we find them in our own experiments?

·        I) The minimum value, for one measurement under ideal conditions, is half the limit of reading. Ex. A ruler with lines 1 mm apart is used to measure a length to 23 mm. Half the limit of reading = 0.5 mm, so the measurement is 23±0.5 mm.

·        II) If we distrust our reading, a higher absolute uncertainty can be estimated. Ex. We take a time of 8.06 s with a stopwatch that measures 1/100 seconds, so half the limit of reading would be 0.005 s. But we know from experience that our reaction time is longer than that, so we estimate it to for example 0.10 s, and have the result 8.06±0.1s.

·        III) If we have several (at least about 5) measurements of the same thing, we can use the highest residual as an absolute uncertainty. A residual = the absolute value of the difference between a reading and the average of the readings.

Ex. Five people measure the mass of an object. The results are 0.56 g, 0.58 g, 0.58 g, 0.55 g, 0.59g.

The average is (0.56g + 0.58g + 0.58g + 0.55g + 0.59g)/5 = 0.572g

The residuals are 0.56g - 0.572g = (-) 0.012g, 0.58g - 0.572g = 0.008g, 0.58g - 0.572g = 0.008g, 0.59g - 0.572g = 0.018g

Then the measurement is m = 0.572g±0.018g or sometimes 0.57±0.02g.

Significant digits

A simpler way of dealing with the issues of uncertainty and error, useful especially in calculations in problem-solving, is to count significant digits. The idea is that the answer should have no more significant digits than the piece of information with the least number of them. As "significant" digits are counted all digits except zeroes in the beginning of a number (ex. 0,00503 has only 3 sig.digs, 5-0-3) and zeroes at the end of an integer (unless we have other reasons to believe they are significant). Ex. 2500 has 2 sigdigs, but can have more if it stands in a table of values from the same source (ex. mass measurements of 2481g, 3113g, 2500g, 4669g etc. - then we can assume that all these have the same precision.)

Errors and graphs

In graphs we can indicate the absolute error with error bars (for the quantity on the horizontal or the vertical axis, or both). Uncertainties in the slope and intercept are the maximum difference between the slopes and intercepts of the best-fit line and those obtained with a maximum fit (within the error bars) and a minimum fit.

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1.6. Orders of magnitude

Finding the order of magnitude

Transform the number to the form X*10^Y , where X is between 0.5 and 5. Then the order of magnitude is 10^Y. Ex. a) 400 = 4 * 10² => 10² b) 600 = 0.6 * 10³ => 10³.

Ranges of magnitudes in the universe

Size: subnuclear particle (proton, neutron) : 10^-15 m, nucleus 10^-14 m, atom 10^-10 m, diameter of earth 10⁷ m, radius of earths orbit 1.5 x 10¹¹m, ....

Mass: mass of electron ca 10^-30 kg , of proton or neutron ca 10^-27 kg. Mass of the planet earth is ca 6*10²⁴ kg or 10²⁵ kg, of the sun ca 10³⁰ kg.

Time: seconds in a year ca 3 * 10⁷, age of universe ca 10 000 million years = 10¹⁰ years or 3 * 10¹⁷ seconds.

Ratios and orders of magnitude

If we know the order of magnitude for two quantities, then we can estimate the order of magnitude for a ratio or other combination of them. For example, if the radius of the earth is ca 10⁷ m, then the volume is ca (10⁷)³ = 10²¹ m³ and consequently the density = mass/volume ca 10²⁵kg/10²¹ m³ = 10⁴ kgm^-3.

Estimating approximative values of everyday quantities

For example, the area of a shoe sole may be 0.1 m * 0.2 m = 0.02 m². The volume of air in a classroom may be 10m * 10m * 2m = 200 m³. Some simplifications will be made here both in the estimated values and in other aspects (the shoe is estimated to have a rectangular area, the classroom to have a box-like shape where the volume of persons and objects in the room is ignored).

1.7. Logarithms

When talking about large numbers, only the order of magnitude is sometimes relevant. If one person has 10 euros, another 1000 euros and a third one 1 000 000 euros then the number of digits or zeroes is more important than the exact value (if you have a few million euros it does not matter whether a thing costs 1000, 2000 or 3000 euros. A mathematical tool for expressing the size of a value are logarithms. A common type of logarithm has the base 10. Examples:

log 10 or often lg 10 = 1, lg 100 = 2, lg 1000 = 3 , lg 1000 000 = 6 etc.

But what is then the logarithm of a number between 100 and 1000? We can see that all numbers between these should have logarithms between 2 and 3. The practical answer is that the calculator gives them (the maths teacher will tell you more about why):

lg 200 = 2.301..., lg 300 = 2.477..., lg 900 = 2.954...

Logarithms can also be made with other bases; one important is the "natural logarithm" or logarithmus naturalis which has the base e = 2.718.... and is written "ln". It would also be useful to have a logarithm with the base 2, but since the applications of this are in "modern" areas like nuclear physics and computer science it did not become common when logartihms were discoverd a few centuries ago. For logarithms there are some special calculation rules, for example:

log (xy) = log x + log y

(think of base 10 logarithms: lg 100000 = 5, 100 000 = 100 * 1000 so for example lg x = 2 and lg y = 3 giving lg 10⁵ = lg 10³ + lg 10²)

Logarithmic graphs will be dealt with more later in the context of radioactive decay.