Dimensions & Units. This discussion (chap 1) is a brief reminder of how to do it.

Transcription

1 Dimensions & Units This course is an introduction to Mechanics, the foundation of Physics. It first builds the basis to express motion in mathematical language (kinematics; chaps 2-4) and then it presents its basic laws (chap 5), e.g. F = ma. In the following chapters (6-13) it explores consequences of these laws. Physics is expressed in mathematical language, therefore we must learn how to work with physical quantities in order for them to appear correctly in equations. This discussion (chap 1) is a brief reminder of how to do it.

2 Units Physical quantities must be measured in order to compare them with predictions and/or other measurements, and this requires UNITS. If you wanted to count fruit, 10 does not specify anything until you specify the units: e.g. pears. In mechanics, the subject of this course, all units can be derived from three basic units: the units of LENGTH, TIME, and MASS. A unit derived from these may, or may not, have its own name. For example, speed is measured in units of length/time (no name), and it can be expressed in terms of any combination of units: ft/s, mi/ hr, km/day, m/s, etc. However, only quantities expressed in like units can be added or subtracted: if a car goes for a mile and skids to a stop over 20 ft, the distanced traveled is 5300 ft or mi. A system of units is one in which LENGTH, TIME, and MASS are expressed in agreed-upon units. In the SI system, they are measured in units of m, s, and kg, respectively. For example force is measured in units of kg m/s 2, which has a special name: Newton; 1kg m/s 2 = 1 N.

3 Units Physics is expressed in mathematical language, therefore we must know how to combine physical quantities algebraically ( +, -, x, / ). For example, if we drop an object, the distance d it travels in a time interval t is d = (1/2)gt 2. What is the time it takes to reach the ground if we drop it 2 ft above it? The time is t = (2d/g) 1/2, but only if we express d and g in the same system of units is the answer meaningful. If we want to use d = 2 ft, we need g = 32.2 ft/s 2 to get t = 0.35 s. If we want to use g = 9.8 m/s 2, we need to convert d to m. In that case d = 0.61 m, and t = 0.35 s. MAKE YOUR LIFE SIMPLE: STICK TO ONE SYSTEM! You can then concentrate on the Physics, and not be confused by how much a meter or a Newton is, or what a Joule or a Watt mean.

4 Converting Units Example: your height (say 5 10 ) in m? ( it s.305 m/ft ) We need the height in ft or in, which we then convert to m. Either way, the height is 1.8 m. Another example: your weight (say 140 lb) in N? ( weight is a force, not a mass. The book gives 1 lb = 4.45 N) The weight corresponds to a mass of 63.6 kg (weight = mass g, g = 9.8 m/s 2 ). We think of weight as mass because g is constant. Lesson: Convert each unit, multiplying or dividing so that unwanted unit cancels out algebraically.

5 Converting Units Lesson: Convert each unit, multiplying or dividing so that unwanted unit cancels out algebraically.

6 Dimensions Physical quantities of the same type (lengths, forces, etc.) have the same dimension, regardless of their units. For example, a distance of 10 ft, an altitude of 1 km, and the radius of the Earth ALL have dimension of LENGTH ( symbol [L] ; [T] & [M] for TIME & MASS ). Dimensions must appear in algebraically correct fashion in physical equations. In other words, if we do the algebra of the dimensions of the quantities appearing in an equation, each term must be of the same dimension. For example, if you forget whether d = (1/2)gt 2 or d = (1/2)t 2 /g, you can check that [d]=[l]=[gt 2 ]=[g][t] 2 because [g]=[l]/[t] 2, while [t 2 /g] is not a length. Checking for dimensional consistency is called dimensional analysis, and it is a rather useful tool in Physics. While not required of you, it can help check answers to problems.

7 Dimensions Simple example: If an object moves with constant acceleration a, starting with velocity v, what can be the form of the equation for the distance traveled? [a]=[l]/[t] 2, while [v]=[l]/[t], so both vt and at 2 have dimension of length. However, so does (vat 3 ) 1/2. Therefore, d = Avt + Bat 2 + C(vat 3 ) 1/2 is dimensionally correct, but A, B, and C are pure numbers (π, ½, etc.) that we cannot obtain from dimensional analysis. One possibility we can exclude, however, is that the correct equation has A=B=0. In that case we would predict that if we drop an object from rest (a=g) d=0! In Chapter 2 we will learn that A=1, B=1/2, and C = 0.

8 Dimensions Famous example: Rayleigh scattering explains why light scattered by a gas of small particles appears blue (hence the blueness of the sky), and it can be understood by dimensional analysis. Light is a wave phenomenon (PHY1012), and light of a given color is characterized by its wavelength λ, the distance between peaks of a sinusoidal wave (for sound, λ corresponds to pitch). Blue light has smaller wavelength than red light. In 1871, John W. Strutt (later Lord Rayleigh) reasoned that the amplitude of light scattered by a particle must be proportional to the amplitude of the incident wave. The particle just redistributes light (neither takes nor adds energy to it). Therefore, the ratio had to be a dimensionless number. Likewise, Rayleigh reasoned, the amplitude of the scattered radiation at distance r away from the particle must be inversely proportional to r (conservation of energy) and proportional to the particle s volume V (dimensional analysis, but you need PHY1012).

9 Dimensions That leaves only the wavelength λ to build the dimensionless ratio. The speed of light or the particle s density cannot appear because they have a dimension (time or mass) that cannot be cancelled out. Therefore, the ratio of amplitudes is given by V/(λ 2 r). The intensity of light (I = energy/area/time) is proportional to the timeaveraged squared amplitude (average of sin( ) is zero, but average of sin 2 ( ) is not ). Hence Rayleigh s law: small particles scatter light with an intensity inversely proportional to the fourth power of the wavelength.

10 Dimensions Rayleigh scattering gives the sky its blue color (with the help of the darkness of outer space, and the color sensitivity of the eye), and the setting sun its orange color.

11 Order of Magnitude Estimation of an order of magnitude is a useful way to check your answer to numerical problems that involve many powers of 10, but it is also a useful tool in physics to know what to expect from a detailed calculation or measurement. Text example: An area of 70 mi 2 (~ 10 8 m 2 ) gets covered by a layer of water.5 in (~ 10 2 m) deep (total volume V ~ 10 6 m 3 ), brought down by N drops of radius r = m [ total drop volume V drop = N(4πr 3 /3) ]. V drop = V for N ~ (we can take π/3 ~ 1 in this kind of calculation).

12 Order of Magnitude (2336/299)2 ~ 60 60x75 = 4,500 penguins

13 The HUDF contains about 10,000 galaxies in this small area and you need about 13,000,000 such squares to cover the entire sky

14 Order of Magnitude You can estimate the thickness of the atmosphere from your own backyard (need clear view near the horizon, where the Sun set). It s easy to estimate 2α. Get the time difference T in hr, and use the fact that the sun moves 360 every 24 hr. Then 2α = T (360 /24hr), from which you get α. Since (R +h)cosα = R, and T ~ 1 hr, h ~ 60 km!

15 Order of Magnitude What s the height of the atmosphere seen in this picture?

16 Order of Magnitude The atmosphere doesn t have an edge. Instead, the air density falls off exponentially with height ( ~ exp[ z km/8 km] ).