Lecture 1.2 Units, Dimensions, Estimations 1. Units To measure a quantity in physics means to compare it with a standard. Since there are many

Transcription

1 Lecture. Units, Dimensions, Estimations. Units To measure a quantity in hysics means to comare it with a standard. Since there are many different quantities in nature, it should be many standards for those quantities. However, there are not as many as you may think. Not all of the hysical quantities are indeendent. For instance, velocity is length dived by time, density is mass dived by volume; volume itself is length multilied by length multilied by length. After all, it turns out that in mechanics there are only indeendent quantities, such as length, time and mass. All other quantities in mechanics can be measured using derived units based on these basic dimensions. So, one has to define standards to measure those quantities. Each standard corresonds to exactly one unit of quantity. However, there is a numerous number of ways how one can define this unit. For instance, we have many units to measure length such as meter, foot, inch, mile and so on. The imortant issue is that scientists around the word have to use the same units, which have to be ractical and sensible. This is why the international system of units (SI) was introduced and now it is used in the most countries of the world. This system is based on the French Metric system. The basic units of this system are Meter (m) for length, Second (s) for time, Kilogram (kg) for mass. The units for other quantities sometimes have their secial names, sometimes they do not, but all of them can be obtained as a result of some sort of combination of basic units. For examle the unit of force has its own name Newton, but Newton= kilogram* meter/ second squared. The unit of momentum does not have the name it is just kg*m/s. The choice of the basic units is dictated by internal roerties of our universe, such as existence of sace, time and matter. These are the basic elements and their roerties have to be described by means of the basic dimensions. Mass is one of the essential but not the only one of the attributes of matter. To measure it we shall use dimension of mass. Continuities of sace and time are measured by means of dimensions of length and time. The choice of the units for these dimensions has a long history. The definition of SI unit of length, meter, was changed several times throughout the history. Now meter is defined as the length of the ath traveled by light in vacuum during the time interval

2 of / of a second. The official system of units in the United States is the British system of units. The basic unit of length in this system is inch=0.054m exactly. The definition of time unit should be based on the time standard, which can be any eriodic henomenon, reeating itself with high accuracy. Based on this, the SI unit of time, second, is now defined as the time it takes for oscillations of light (with secified wavelength) emitted by Cesium- atom. The SI unit of mass is just the mass of the secial latinum-iridium cylinder ket in International Bureau of Weights and Measures in Sevres (France). Since there are different systems of units, it is clear that one often needs to convert units from one system to another. This can be done with the hel of conversion factors (a ratio of units that is equal to unity). Different useful conversion factors can be found in Aendix D of the book. Examle. Convert a velocity of 60mh to SI units. Work this examle through before reading the solution. Since the SI units for length are meters and for time are seconds, we have to convert this velocity to m/s. To do so, we have to use several conversion factors. First let us notice that mi=580ft, ft=in, in=0.054m, h=60min and min=60s. So we have mi mi 580 ft 580 ft 60mh h h 60 min 60 min 580 in 580 in m m s s s s The book rovides some other examles of unit conversion. Please take a look at Samle Problem in Chater of the book.. Dimensions Every hysical quantity has a certain dimension. In articular, we have secified three basic dimensions existing in classical mechanics: length, time and mass. As we just have seen these quantities can be measured using different systems of units. However, regardless of the system of units, each articular hysical quantity still has its own dimension. For examle, length is always measured in units of length. These can be either meters or feet or some other units of length but not units of time such as seconds. Other hysical quantities may have more comlicated dimensions. For instance, the dimension of velocity is dimension of length dived by time. So, velocity should always have this dimension in any system of units, it could be meters er second or miles er hour, but always dimension of length divided by dimension of time.

3 This means that any valid formula in hysics should be dimensionally consistent. For instance, it does not make sense to add length and time, because these two quantities have different dimensions. So, the result of such summation does not have any hysical sense. On the other hand, it makes sense to divide length by time, because the result of this mathematical oeration has dimension of velocity and can be interreted as such. When solving hysics roblems, you should always check whether or not your answer has the right dimension, as well as that your equations are dimensionally consistent. It is convenient to introduce the secial notation for dimensions, using square brackets. For instance [l] means dimension of length and [t] means dimension of time. Then velocity has dimension [l]/[t]=[l/t], and acceleration, which is the change of velocity er unit of time has dimension of [l]/[t]/[t], which is lt. Examle. Check dimensional consistency of a simle equation x x v t, where x and x 0 reresent distances, v 0 velocity and t time. Work through this examle before reading the solution. Dimensions for both sides of this equation are [ l [ ] ] [ l l ] [ ] [ t ] [ l t ] [ l ] [ l ]. 0 0 This means that both sides of this equation have the same dimension and therefore dimensionally consistent. In many hysics roblems it is necessary to obtain the answer in terms of the given variables. Even if you do not know the exact equation to relate the unknown with those variables, there is a owerful method which allows you to, at least, obtain the functional relation between the variables. This method is known as Dimensional Analysis. It is based on the knowledge of dimensions for different quantities involved in the roblem. Examle. The roblem asks to find velocity of the car, which starts from rest and moves with acceleration a travelling for distance x. We have not studied the relation between these quantities yet. But I have already mentioned that velocity has dimension of length divided by time [l]/[t], distance has dimension of length [l] and acceleration has dimension of length divided by time squared [ l]/[ t ]. We can make an assumtion that velocity has the form q v a x, here and q are some unknown owers. Using this assumtion try to find and q before reading the solution.

4 Dimension of this relation then will be a x q q. F l q l l H G [ ] I [ ] t K J [ ] [ ] [ t ] On the other hand this relation should have dimension of velocity, so q. [ l] [ l ] [ t] [ t ] Comaring owers of l and t on both sides of this equation, one can see that +q= and =, so =/ and q=-=-/=/. After all we have, v a x v ax. So, velocity is roortional to the square root of acceleration multilied by distance. Even though we cannot find the coefficient of roortionality, using dimensional analysis, but it could be obtained from somewhere else, for instance, from exeriment.. Order-of-magnitude Estimations Estimation is a useful technique for getting an aroximate answer and for determining whether or not the answer you are getting is reasonable. If you solve a roblem in hysics always check that your answer is reasonable. For instance, the distance from the Earth to the Sun cannot be of the same order of magnitude as the distance from one building on McMurry camus to another building there. Since hysicists use to work with scientific notation, they like to use Order of magnitude estimations. This means that the number is correct to within a factor of ten. We also use the large amount of aroximations when solving roblems in hysics. In certain cases one can ignore air resistance when solving roblem about rojectile moving near the earth s surface. Sometimes we can ignore the shae of the hysical body relacing it by something which has similar but more erfect shae, so we can do reasonable estimations. Whatever aroximation we are using to solve the roblem, we have to remember that certain aroximations can only work at certain conditions. We always have to make sure that these conditions are valid in each articular situation. We should also be able to make redictions of how the answer for the roblem may change if we are not using the aroximation any longer.

5 Examle.4 You are working for a radio station. The general manager wants to do a romotional stunt: if her favorite football team goes undefeated, she wants to fill the stadium with Oreo cookies. How much would it cost to do this? Make your own estimation before looking at the solution. First, estimate the length, width and height of the stadium. We ll aroximate the stadium as a box. The field is 00 yards long, lus about 0 yards for each end zone, and some extra. Let s say about 50 all together l =.5 x 0 yards The width is about 00 yards (40 yards for the field, lus sidelines) The height is about 0 yards So the volume of the stadium is: w =.00 x 0 yards h=.0 x 0 yards. V = l w h =.5 x 0.0 x 0.0 x 0 = (.5.0.0) x 0 (++) ( ) =.0 x 0 5 Oreo cookies are about in diameter and ½ thick. ft r = = in ft in ft h = 6 in ft in 7 4 ft ft V oreo = r h= = x0 4x0 7x0 = 8x0 =.8x0 4 =

6 V Oreo = 0 x 4 How many cookies fit in the stadium? # cookies = V stadium V Oreo. 0x0 x ( 4) = 0 x0 = x = 0 0 x cookies About 0 billion Oreos could fit inside the stadium. If you figure an average bag has about 00 Oreos for $.00, this stunt would cost. x0 0 $. 00 cookies x0 x0 cookies 8 dollars! It is now time to try things by yourself