Physics 121 9/3/10 Prof. E. F. Redish 1 1 •Theme Music: Paul Simon When numbers get serious •Cartoon: Bill Waterson Calvin & Hobbes 9/3/10 Math is different in science from in math class! • Math in physics is fundamentally about relationships among measurements, not primarily about calculating – or even solving (though we do some of that). • Functional dependence is critical. “More of this means more of that” is not good enough. How much more matters. 9/3/10 7 An example from a math exam • Writing the equation in this problem on a physics exam would receive 0 credit and the comment: “This is a meaningless equation!” The population density of trout in a stream is where r is measured in trout per mile and x is measured in miles. x runs from 0 to 10. (a) Write an expression for the total number of trout in the stream. Do not compute it. 9/3/10 8 r(x) = 201+ xx2+1How would you fix this?Physics 121 9/3/10 Prof. E. F. Redish 2 Third icon: Measurement • As we learned with our example, often our perceptions play us false. • In order to be sure we understand what is going on we need to quantify. In order to do that we need to invent a process to assign a number to a physical quantity – measure. 9 9/3/10 Measurement is basically about counting. 9/3/10 10 N Perim Area Vol 1 2 3 11 Operational definitions •In science we often define something by telling how it is measured, • To assign a number to anything you need – an idea about the character of the object – a process for assigning the number – a scale (usually arbitrary) • In the case of our cubes what scale did we choose? 9/3/10Physics 121 9/3/10 Prof. E. F. Redish 3 12 Dimensions • For every new arbitrary scale we choose, we assign a dimension. – A dimension specifies the kind of measurement (or combination of measurements) we are measuring to get the number. – The dimension we have just defined is LENGTH • Dimensions are useful – for inventing new equations – for catching errors in math – for understanding how a quantity will change when its scale changes. “Unit” specifies the particular scale we have chosen to measure with. 9/3/10 13 Examples • In the next week or so, we will discuss measurements of – length (L) – time (T) – mass (M) • We write the dimensions of a combined quantity like this. If v is a symbol representing velocty that is constructed by dividing a length measure by a time measure, we write v = 87 km/hr [v] = L/T 9/3/10 Measuring Length • The first basic concept we are going to quantify is length. • Propose a way of defining it a process for measuring it. – Scale? – Arbitrariness? – How “real” is this? 9/3/10 14Physics 121 9/3/10 Prof. E. F. Redish 4 Measuring Position • What’s the difference between length and position? • What do you need to measure position? 9/3/10 15 16 Dimensional Analysis and Scaling • Since the measurement scale for a dimension is arbitrary, we could change it and the number assigned to a physical length would change. • A dimensional analysis tells us how a quantity changes when the measurement scale is changed. • Any equation which is supposed to represent a physical relation must retain its equality when we make a different choice of scale. • Dimensional analysis tells us how something changes when we change our arbitrary scale. 9/3/10 Dimensioned quantities are not just numbers 17 1 inch = 2.54 cm1 = 2.54 cm1 inchYou can multiply anything by 1 without changing it! 12 inches = (12 inches) 1 = (12 inches) 2.54 cm1 inch = 12 2.54 cm = 30.48 cm9/3/10Physics 121 9/3/10 Prof. E. F. Redish 5 Changing units •To change units, we use the fact that the unit clings to a number like a multiplied symbol in algebra, 6a or 3b. • We then manipulate our units like symbols. 18 1 hour = 60 minutes1 = 60 minutes1 hour16mimin=16mimin 1=16mimin60 min1 hour=606(mi)(min)(min)(hour)= 10mihour9/3/10 The unit is part of the expression. 19 1 m = 100 cm1 m()2= 100 cm()21 m2= (102)2 cm2= 104 cm210 cm is what part of a meter? (10 cm)x(10 cm) = 100 cm2 is what part of a meter2? 9/3/10 20 Careful! •Dimensions are not algebraic symbols – they are type labels. 6 ft + 9 ft = 15 ft [6 ft] + [9 ft] = [15 ft] L + L = L • We sometimes use “L” (or “M” or “T”) for algebraic symbols – to specify a particular length or mass or time. You have to know whether you are doing a dimensional analysis or a calculation! 9/3/10Physics 121 9/3/10 Prof. E. F. Redish 6 An example •If I run a 6 minute mile, what is my speed? • “6 minute mile” is a pace p = (time)/(distance). • Speed is the upside-down pace v = (distance)/(time) • Therefore, v = 1/p 21 p =6 min()(1 mi)v =1p=16 min1 mi()=1 mi6 min=16mimin9/3/10 22 What have we learned? •In physics we have different kinds of quantities depending on how they were measured. • These quantities change in different ways when you change your measuring units. • Only quantities of the same type may be equated (or added) otherwise an equality for one person would not hold for another. • Measurements are not numbers. They represent physical quantities and therefore contain units as part of them. 333cm 5cm 4cm 1 =+)(anythings 5cm 4 cm 12+ 9/3/10 23 Letting dimensional analysis work for you • In physics, if we try to add or equate quantities of different dimensions we get nonsense. • If we didn’t maintain dimensional correctness, an equality that worked in one measurement system wouldn’t work in another. • This is a very good way to check your work with equations. (But it’s hard to do if you put numbers in too early!*) 9/3/10 * You also won’t get much partial credit on exams if you put numbers in too early since we may not be able to tell what equations you are using and
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