Chapter 1: Units, Physical Quantities and Vectors

1. Chapter 1: Units, Physical Quantities and Vectors

2. About Physics

3. What is Physics? • Phys’ics[Gr. Physika, physical or natural things] • Originally, natural sciences or natural philosophy • The science of dealing with properties, changes, interaction, etc., of matter and energy • Physics is subdivided into mechanics, thermodynamics, optics, acoustics, etc. • From Webster's Unabridged Dictionary

4. Science • Science[Latin scientia - knowledge] • Originally, state of fact of knowing; knowledge, often as opposed to intuition, belief, etc. • Systematized knowledge derived from observation, study and experimentation carried on in order to determine the nature or principles of what is being studied. • A Science must have PREDICTIVE power

5. Physics: Like a Mystery Story • Nature presents the clues • Experiments • We devise the hypothesis • Theory • A hypothesis predicts other facts that can be checked - is the theory right? • Right - keep checking • Wrong - develop a new theory • Physics is an experimental science

6. The Ancient Greeks Aristotle (384-322 B.C.) is regarded as the first person to attempt physics, and actually gave physics its name. On the nature of matter: Matter was composed of: Air Earth Water Fire Every compound was a mixture of these elements Unfortunately there is no predictive power

7. On the Nature of Motion • Natural motion - like a falling body • Objects seek their natural place • Heavy objects fall fast • Light objects fall slow • Objects fall at a constant speed • Unnatural motion - like a cart being pushed • The moving body comes to a stand still when the force pushing it along no longer acts • The natural state of a body is at rest

8. Aristotelian Physics • Aristotelian Physics was based on logic • It provided a framework for understanding nature • It was logically consistent It was wrong !!! • Aristotelian physics relied on logic - not experiment

9. Observe Abstract Hypothesis Prediction Experiment The Renaissance Galileo Galilei (1564 -1642) was one of the first to use the scientific method of observation and experimentation. He laid the groundwork for modern science.

10. Classical Mechanics Newton’s Laws work fine for • Large Objects - Ball’s, planes, planets, ... • Small objects (atoms)  Quantum Mechanics • Slow Objects - people, cars, planes, ... • Fast objects (near the speed of light)  Relativity • Classical Mechanics - essentially complete at the end of the 19th Century Mechanics: the study of motion Galileo (1564 -1642) laid the groundwork for Mechanics Newton (1642-1727) completed its development (~almost~)

11. Why is Physics Important? • Planetary motion • Steam Engines • Radio • Cars • Television Newton’s Laws andClassical Physics QuantumMechanics The NextGreat Theory • Microwaves • Transistors • Computers • Lasers • Teleportation • Faster than light travel(can’t exist today) "Heavier-than-air flying machines are impossible." Lord Kelvin, president, Royal Society, 1895.

12. Mechanics • Physics is science of measurements • Mechanics deals with the motion of objects • What specifies the motion? • Where is it located? • When was it there? • How fast is it moving? • Before we can answer these questions • We must develop a common language

13. Units

14. Fundamental Units Foot Meter - Accepted Unit Furlong Length [L] Second - Accepted Unit Minute Hour Century Time [T] Kilogram - Accepted Unit Slug Mass [M]

15. Derived Units • Single Fundamental Unit • Area = Length  Length [L]2 • Volume = Length  Length  Length [L]3 • Combination of Units • Velocity = Length / Time [L/T] • Acceleration = Length / (Time  Time) [L/T2] • Jerk = Length / (Time  Time  Time) [L/T3] • Force = Mass  Length / (Time  Time) [M L/T2]

16. Units • SI(Système Internationale)Units: • mks:L = meters (m), M = kilograms (kg), T = seconds (s) • cgs:L = centimeters (cm), M = grams (g), T = seconds (s) • British Units: • Inches, feet, miles, pounds, slugs... • We will switch back and forth in stating problems.

17. Unit Conversion • Useful Conversion Factors: • 1 inch = 2.54 cm • 1 m = 3.28 ft • 1 mile = 5280 ft • 1 mile = 1.61 km • Example: convert miles per hour to meters per second:

18. Orders of Magnitude • Physical quantities span an immense range • Length size of nucleus ~ 10-15 m size of universe ~ 1030 m • Time nuclear vibration ~ 10-20 s age of universe ~ 1018 s • Mass electron ~ 10-30 kg universe ~ 1028 kg

19. Physical Scale • Orders of Magnitude Set the Scale • Atomic Physics ~ 10-10 m • Basketball ~ 10 m • Planetary Motion ~ 1010 m • Knowing the scale lets us guess the Result Q: What is the speed of a 747? Distance - New York to LA 4000 mi = 660 mph Flying Time 6 hrs

20. Dimensional Analysis • Fundamental Quantities • Length - [L] • Time - [T] • Mass - [M] • Derived Quantities • Velocity - [L]/[T] • Density - [M]/[L]3 • Energy - [M][L]2/[T]2

21. Physical Quantities • Must always have dimensions • Can only compare quantities with the same dimensions • v = v(0) + a  t • [L]/[T] = [L]/[T] + [L]/[T]2 [T] • Comparing quantities with differentdimensions is nonsense • v = a  t2 • [L]/[T] = [L]/[T]2 [T]2 = [L]

22. Provides Solution Sometimes • Period of a Pendulum Which of these could be correct? Period is a time [T] - t Can only depend on: Length [L] - l Mass [M] - m Gravity [L/T2] - g

23. Solving Problems

24. Problem Solving Strategy • Each profession has its own specialized knowledge and patterns of thought. • The knowledge and thought processes that you use in each of the steps will depend on the discipline in which you operate. • Taking into account the specific nature of physics, we choose to label and interpret the five steps of the general problem solving strategy as follows:

25. A. Everyday language:  1) Make a sketch.   2) What do you want to find out?   3) What are the physics ideas? B. Physics description:  1) Make a physics diagram.   2) Define your variables.   3) Write down general equations. D. Calculate solution:  1) Plug in numerical values. E. Evaluate the answer:  1) Is it  properly stated?   2) Is it reasonable?   3) Answered the question asked? Problem Solving Strategy • C. Combine equations:  1) Select an equation with the target variable.   2) Which of the variables are not known?   3) Substitute in a different equation.   4) Continue for all of the unknown variables .   5) Solve for the target variable.   6) Check units.

26. Problem Solving Strategy, Step A A. Everyday language description: In this step you develop a qualitative description of the problem. • Visualize the events described in the problem by making a sketch.  The sketch should indicate the different objects involved and any changes in the situation (e.g. changes in force applied, collisions, etc.)  First, identify the different objects that are relevant to finding your desired category.  Next, identify whether there is more than one stage (part) to the behavior of the object during the time from the beginning to the end that is relevant for what you are trying to find out.  Things that would indicate more than one part would include key information about the behavior of the object at a point between start and end of movement, collisions, changes in the force applied or acceleration of an object. • Write down a simple statement of what you want to find out.  This should be a specific physical quantity that you could calculate to answer the original question. • Write down verbal descriptions of the physics ideas (the type of problem). Identify the physics idea for each stage of each object. If the physics idea is a vector quantity (motion, force, momentum, etc.) identify how many dimensions are involved.

27. Problem Solving Strategy, Step B B. Physics description: • In this step you use your qualitative understanding of the problem to prepare for the quantitative solution. • First, simplify the problem situation by describing it with a diagram in terms of simple physical objects and essential physical quantities. Make a physics diagram. You will need a diagram for each physics idea for each object, and possibly one for each stage and for each dimension. • Define your variables (make a chart) of know quantities and unknown quantities.  Identify the variable you will solve for.  Make sure variables are defined for each object, stage, idea and dimension.  Pay attention to units, to make sure you have the right kind of units for each type of variable. • Using the physics ideas assembled in A-3 and the diagram you made in B-1, write down general equations which specify how these physical quantities are related according to the principles of physics or mathematics. 

28. Problem Solving Strategy, Step C C. Combine equations: • In this step you translate the physics description into a set of equations which represent the problem mathematically by using the equations assembled in step 2. • Select an equation from the list in B3 that contains the variable you are solving for (as specified in B2). • Identify which of the variables in the selected equation are not known. • For each of the unknown variables, select another equation from the list in B3 and solve it for the unknown variable.  Then substitute the new equation in for the unknown quantity in the original equation. • Continue steps 2 & 3 until all of the unknown variables (except the variable you are solving for) have been replaced or eliminated. • Solve for the target variable. • Check your work by making sure the units work out.

29. Problem Solving Strategy, Steps D & E D. Calculate solution: • In this step you actually execute the solution you have planned. • Plug in numerical values (with units) into your solution from C-5. E. Evaluate the answer: • Finally, check your work. • Is it  properly stated? Is it reasonable? • Have you actually answered the question asked?

30. Problem Solving Strategy • Consider each step as a translation of the previous step into a slightly different language. • You begin with the full complexity of real objects interacting in the real world and through a series of steps arrive at a simple and precise mathematical expression. The five-step strategy represents an effective way to organize your thinking to produce a solution based on your best understanding of physics. The quality of the solution depends on the knowledge that you use in obtaining the solution. • Your use of the strategy also makes it easier to look back through your solution to check for incorrect knowledge and assumptions. That makes it an important tool for learning physics. • If you learn to use the strategy effectively, you will find it a valuable tool to use for solving new and complex problems.

31. Vectors

32. A scalar is a physical quantity that has only magnitude (size) and can be represented by a number and a unit. Examples of scalars? Time Mass Temperature Density Electric charge A vector is a physical quantity that has both magnitude (size) and direction. Examples of vectors? Velocity Force Scalars & Vectors

33. Displacement Vectoris a change in position. It is calculated as the final position minus the initial position. • Vectors are • represented pictorially by an arrow from one point to another. • represented symbolically by a letter with an arrow above it.

34. Two vectors that have the same direction are said to be parallel. Two vectors that have opposite directions are said to be anti-parallel. Two vectors that have the same length and the same direction are said to be equal no matter where they are located. The negative of a vector is a vector with the same magnitude (size) but opposite direction Some Vector Properties

35. Magnitude of a Vector • The magnitude of a vector is a positive number (with units!) that describes its size. • Example: magnitude of a displacement vector is its length. • The magnitude of a velocity vector is often called speed. • The magnitude of a vector is expressed using the same letter as the vector but without the arrow on top of it.

36. Vector Addition • Vector C of a vector sum of vectors A and C. • Example: double displacement of particle. • Vector addition is commutative (the order of vector addition doesn’t matter).

37. Vector Addition C A U T I O N • Common error: to conclude that if C = A + B the magnitude C should be equal the magnitude A plus magnitude B. Wrong ! • Example: C < A + B.

38. Vector Addition • Add more than two vectors:

39. Vector Subtraction • Subtract vectors:

40. Vector Components • There are two methods of vector addition • Graphical  represent vectors as scaled-directed line segments; attach tail to head • Analytical  resolve vectors into x and y components; add components

41. Vector Components

42. Vector Components • If Rx< 0 and Ry > 0 or if Rx< 0 and Ry < 0 then  + 180o

43. Vector Components C A U T I O N • The components Axand Ay of a vector A are numbers; they are not vectors !

44. Vector Components

45. Vector Components

46. IDENTIFYthe relevant concepts andSET UP the problem: Decide what your target variable is. It may be the magnitude of the vector sum, the direction, or both. Then draw the individual vectors being summed and the coordinate axes being used. In your drawing, place the tail of the first vector at the origin of coordinates; place the tail of the second vector at the head of the first vector; and so on. Draw the vector sum R from the tail of the first vector to the head of the last vector. By examining your drawing, make a rough estimate of the magnitude and direction of R you’ll use these estimates later to check your calculations. Problem Solving Strategy VECTOR ADDITION

47. Vector Components • There are two methods of vector addition • Graphical represent vectors as scaled-directed line segments; attach tail to head • Analytical resolve vectors into x and y components; add components Component vectors Components

48. Vector Components • You can calculate components if its magnitude and direction are known • Direction of a vector described by its angle relative to reference direction • Reference direction  positive x-axis • Angle the angle between vector A and positive x-axis y 90 < Θ< 180 cos (-) sin (+) 0 < Θ< 90 cos (+) sin (+) Θ= 90 Θ= 180 Θ= 0 x 180 < Θ< 270 cos (-) sin (-) 270 < Θ< 360 cos (+) sin (-) Θ= 270

49. Vector Components

50. Vector Components C A U T I O N • The components Axand Ay of a vector A are numbers; they are not vectors ! • The components of vectors can be negative or positive numbers. 90 < Θ< 180 cos (-) sin (+) 180 < Θ< 270 cos (-) sin (-)