PHYSICS 149: Lecture 2 Chapter 1 1.1 Why study physics? 1.2 Talking physics 1.3 The Use of Mathematics 1.4 Scientific Notation and Significant Figures 15Units 1.5 1.6 Dimensional Analysis 1.7 Problem-Solving Techniques 1.8 Approximation 1.9 Graphs (self study) Lecture 2 Purdue University, Physics 149 1
Equations Understand the concepts first! Equations allow us to use the concepts to make predictions quantitatively. Physicists perform experiments to verify predictions. Units are an essential part of each value in equations and must be handled using dimensional analysis techniques Vector equations relate direction as well as magnitude. You will learn about vectors in this course. The car moved 2 miles The car moved 2 miles west provides more information West Lecture 2 Purdue University, Physics 149 2
Significant Figures The result of a calculation indicates the precision of the measurement with the significant figures. The number of significant figures in a measurement, such as 2.531 g, is equal to the number of digits that are known with some degree of confidence (2, 5, and 3) plus the last digit (1), which is an estimate or approximation. Postage Scale 3 (1 g) 1 significant figure Two-pan balance 2.53 (0.01 g) 3 significant figures Analytical balance 2.531 (0.001 g) 4 significant figures Zeros within a number are always significant. Both 4308 and 40.05 contain four significant figures. Zeros that do nothing but set the decimal point are not significant. Thus, 470,000 has two significant figures. Trailing zeros that aren't needed to hold the decimal point are significant. For example, 4.00 has three significant figures. Lecture 2 Purdue University, Physics 149 3
Significant Figures Significant figures tell the precision of a measurement. 12 g is not the same as 12.0 g in terms of precision. 12 g implies a value more than 11 and less than 13. 12.0 g tells you the value is between 11.9 and 12.1 All non-zero digits are always significant: 1.23 12.3 123 all have 3 sig. figs Leading zeros are never significant: 0.00130013 = 2 sig. figs 014 0.14 = 2 sig. figs Trapped zeros are always significant: 1.002 = 4 sig. fig Trailing zeros are significant if they are after a decimal point and if they are after a non-zero digit: 12.00 = 4 sig. fig 0.00120 = 3 sig. fig (the first three zeros are leading ) Lecture 2 Purdue University, Physics 149 4
Units To communicate the result of a measurement for a quantity, a unit must be defined! Defining units allows everyone to relate to the same fundamental amount Always write down units and carry the units through all of the calculations Prefix (abbreviation) Power of Ten Peta (P) 10 15 Tera (T) 10 12 Giga (G) 10 9 Mega (M) 10 6 Kilo (k) 10 3 We use SI system 1 meter = 3.281 ft Hecto (h) 10 2 Deci (d) 10-1 Length: meters Mass: kilograms Time: seconds 1 pound = 4.448 N 1 kg weighs 2.205 pounds (where g = 32.174ft/s 2 ) Dimensional Analysis: Both sides of an equation must have the same dimensions Can be used to verify equations, answers Centi (c) 10-2 Milli (m) 10-3 Micro (μ) 10-6 Nano (n) 10-9 Pico (p) 10-12 Femto (f) 10-15 Lecture 2 Purdue University, Physics 149 5
The Power of 10 General rules about power of 10 calculations: 1 = 10 n 10 n 10 n 10 m = 10 n+m 10 n = 10 n m 10 m 10 Lecture 2 Purdue University, Physics 149 6
SI Units We will use the SI Units (Systeme International d Unites) not the English Units. The SI system is often called the metric system. In the SI systems, there are seven base units including the meter (m), kilogram (kg), and second (s). (See Table 1.1 at page 8). Derived units are constructed from combinations of the base units. For example, the SI unit of force is kg m/s 2. It can also be written as N (Newton) that is a derived unit. (See the inside front cover of your textbook for a complete listing.) Lecture 2 Purdue University, Physics 149 7
SI Units The standard kilogram is a cylinder of metal composed of an alloy of platinum-iridium kept at Sèvres, France. Atomic clock: natural vibrations of cesium atoms (9.2 billion ticks per second) NIST-F1 Cesium Fountain Atomic Clock: The Primary Time and Frequency Standard for the United States Lecture 2 Purdue University, Physics 149 8
SI Units: Meter 1799: one ten-millionth part of the quadrant of the earth based on a measurement of a meridian between Dunkirk and Barcelona. A platinum bar at a specified temperature was the standard for most of the 19th century. 1983: the meter is the length of the path traveled by light in vacuum during a time interval of 1/299,792,458 of a second. The speed of light is c = 299,792,458 m/s The goal was to improve the precision of the definition, and change its length as little as possible. Scientists are now working to redefine the kg from atomic level definition All other variables can be defined in terms of m, Kg and s Example 1N = 1 Newton = unit of force = kg. m/s 2 Lecture 2 Purdue University, Physics 149 9
Conversion of Units Often problem do not give us quantities in SI units and we must convert to SI units. The interstate speed limit is 70 miles/hours. Let us try to convert it to meters/second. Use: 1 mile = 1.609 km = 1609 m 1 hour = 60 min = 3600 s =1 =1 miles miles 1609m 1hour speed = 70 = 70 hours hours 1 miles 3600s speed = 31m / s Lecture 2 Purdue University, Physics 149 10
ILQ: Units A very good fastball pitcher can throw the ball 100 mph. What is the ball speed in m/s? A) 224 m/s B) 44.7 m/s C) 0.0444 m/s D) None of the above miles miles 1609m 1hour speed = 100 = 100 hours hours 1miles 3600s speed = 44.7m / s Lecture 2 Purdue University, Physics 149 11
Example A blue whale has a mass of 1.9 10 5 kg. Assume the average density is ρ=0.85 g/cm 3. What is is the volume of the whale. V = M = 1.9 105 Kg = 1.9 105 Kg 1000g ρ 0.85g / cm 3 0.85g / cm 3 1Kg 1m 10 2 cm 1.9 10 5 Kg 10 3 g 1 m 3 = 105 3 6 3 = 0.85g / cm 3 1Kg 10 6 cm 3 2.23 10 10 m 2.23 10 2 m 3 Checkmarks: - Conversion of units - Correct unit at the end Lecture 2 Purdue University, Physics 149 12 3 =
Conversion Strategies 1. In all calculations write down the units explicitly from the beginning. g 2. Treat all units as algebraic quantities. For example, when identical units are divided they are eliminated algebraically. 3. Use conversion factors located on the back of the front cover. 4. In your calculation recall that multiplying and dividing by 1 does not alter an equation. 5. Check that your calculation is correct by verifying that you get the correct unit for the answer. 6. Only quantities with the same units can be added or subtracted. Lecture 2 Purdue University, Physics 149 13
ILQ Q: A kilometer is approximately A) 1/10 mile B) 2 miles C) 1/2 mile D) 1/4 mile 1 km = 0.6214 mi E) 1/6 mile Lecture 2 Purdue University, Physics 149 14
Dimensional Analysis Q: What is for? A: To check mathematical relations for the consistency of their dimension Example: A car starts from rest and accelerates to a speed v in a time t. You want to calculate the distance but you can not remember if the correct formula is: x = 1 vt 2 or x = 1 2 2 vt Lecture 2 Purdue University, Physics 149 15
Dimensional Analysis You can use dimensional analysis x = 1 vt 2 2 L L = T 2 T = L T x = 1 2 vt L = L T T T = L Definitely wrong You can only state that x is proportional to vt. Limitation: The factor ½ might be wrong! Lecture 2 Purdue University, Physics 149 16
Approximations Q: Why is a rough estimate often useful? A) Simplified models can help us to understand more complex situations B) It might be impossible to measure precisely a quantity needed C) You might need to use mathematical approximations to solve a problem Lecture 2 Purdue University, Physics 149 17
Approximations We will use several mathematical approximations: 1) Small angle approximation sin θ θ cos θ 1 tan θ θ 2) Binomial approximation. If x is small (1+x) n 1+nx (1-x) n 1-nx (1+x) 1/2 1+x/2 Lecture 2 Purdue University, Physics 149 18
Problem Solving Read the problem Read the problem again! Draw a sketch Organize the given information Identify the goal Plan your approach -- the physics is here! Think about the concepts Break the problem into parts Write an equation for each part Perform the algebra Substitute values with units Check units Calculate result Is the result reasonable? Lecture 2 Purdue University, Physics 149 19
Trigonometry θ c a b c 2 = a 2 + b 2 sin θ = b/c cos θ = a/c tan θ = b/a θ Full circle: 0 0 to 360 0 Lecture 2 Purdue University, Physics 149 0 to 2π radians 20
Using Trigonometric Functions A lake drops at an angle θ and we need to know the depth at 22 m. You know that the depth at 14 m is 2.25 m. 1) Find θ 1 =14m =22m tanθ = θ 1 d 1 =2.25m d d 1 d 1 1 θ = tan l 1 2.25 θ = tan 1 14 = 9.130 l 1 Even better: d d = l tanθ = l d 1 l 1 = 22 2.25 14 = 3.54m 2) Find d d = l tanθ d = ( 22m ) ( tan9.13 0 )= 3.54m Lecture 2 Purdue University, Physics 149 21
ILQ Q: What is the number of cars in the US? A) 100,000 000 B) 1,000,000 C) 100,000,000 D) 10,000,000 Example: 1) Find out the number of household h in US: 107M 2) Determine the average number of cars/households: 1.9 3) The total number of cars in the US: 204,000,000 Lecture 2 Purdue University, Physics 149 22
Force Force: a push or pull that t one object exerts on another. Force can also be defined as any action that alters a body s state of rest or of constant speed motion in a straight line. Forces always exist in pairs. Anything that exerts a force also has a force exerted on it. For example, when you push on the ground to walk, the ground pushes back on your foot. Two types of forces on macroscopic objects: Long-range forces: Forces that do not require the two objects to be touching. Example:) gravity and electromagnetic forces. Contact forces: Forces that exist only as long as the objects are touching one another. Example: kicking a ball. Measuring forces: The SI unit for force is the newton (N). One way to measure forces is using springs (F x). Force is a vector (not a scalar). Lecture 2 Purdue University, Physics 149 23
Vector vs. Scalar Vector: a quantity with both (a) magnitude and (b) direction Velocity, weight, force, and so on. Vectors have their own addition/subtraction rules: Do not just add or subtract their magnitudes. In this lecture (unlike your textbook), a vector will be often indicated d by a boldface symbol without t an arrow. Eample: v, W, and F. Scalar: a quantity with magnitude only Speed, mass, temperature, and so on. Scalars can be added or subtracted arithmetically. Lecture 2 Purdue University, Physics 149 24
Net Force The net force is the vector sum of all the forces acting on an object. F 1 F 2 F = 10 N 1 = 10 N = 10 N = 12 N East West East West F 2 If the book is at rest, it will continue to stay at rest because F net =0 (translational equilibrium). If the book is at rest, it will start moving toward the west since F 2 > F 1 (F net 0). Lecture 2 Purdue University, Physics 149 25
Adding Vectors in 1-D If two vectors are in the same direction, their sum is in the same direction and its magnitude is the sum for the magnitudes of the two. If two vectors are in opposite directions, the magnitude of their sum is the difference between een the magnitudes of the two vectors, and the direction of the vector sum is the direction of the larger of the two. Lecture 2 Purdue University, Physics 149 26