Thursday, June 7, 2012PHYS , Summer 2012 Dr. Jaehoon Yu 1 PHYS 1443 – Section 001 Lecture #2 Thursday, June 7, 2012 Dr. Jaehoon Yu Chapter 1 Standards and units Estimates Dimensions and dimensional analysis One Dimensional Motion Fundamentals in Kinematics Displacement Average Speed and Velocity Instantaneous Speed and Velocity Acceleration Today’s homework is homework #2, due 10pm, Thursday, June 14!!

Thursday, June 7, PHYS , Summer 2012 Dr. Jaehoon Yu 2 Announcements Reminder for Reading assignment #1: Read and follow through all sections in appendices A and B by Monday, June 11 –There will be a quiz coming Tuesday, June 12, on this reading assignment Homework –19 out of 21 registered so far... Excellent!! The problem with Quest submission system has been resolved. Please pick any answer and submit. You don’t have to get it right! Four have submitted their answers! –Some homework tips When inputting answers to the Quest homework system –Unless the problem explicitly asks for significant figures, input as many digits as you can –The Quest is dumb. So it does not know about anything other than numbers More details are The first term exam is on Thursday, June 21 –Will cover CH1 – what we finish on Tuesday, June 19 + Appendices A and B –Mixture of multiple choices and free response problems

Thursday, June 7, PHYS , Summer 2012 Dr. Jaehoon Yu 3 Needs for Standards and Units Seven fundamental quantities for physical measurements –Length, Mass, Time, Electric Current, Temperature, the Amount of substance and the Luminous intensity Need a language that everyone can understand each other –Consistency is crucial for physical measurements –The same quantity measured by one must be comprehendible and reproducible by others –Practical matters contribute A system of unit called SI SI ( System Internationale ) was established in 1960 – Length – Length in meters (m)(m) – Mass – Mass in kilo-grams ( kg ) – Time – Time in seconds (s)(s)

Thursday, June 7, PHYS , Summer 2012 Dr. Jaehoon Yu 4 Definition of Three Relevant Base Units One second is the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the Cesium 133 (C 133 ) atom. 1 s (Time) It is equal to the mass of the international prototype of the kilogram, made of platinum-iridium in International Bureau of Weights and Measure in France. 1 kg (Mass) = 1000 g One meter is the length of the path traveled by light in vacuum during the time interval of 1/299,792,458 of a second. 1 m (Length) = 100 cm DefinitionsSI Units There are total of seven base quantities (see table 1-5 on page 7) There are prefixes that scales the units larger or smaller for convenience (see pg. 7) Units for other quantities, such as Newtons for force and Joule for energy, for ease of use

Thursday, June 7, PHYS , Summer 2012 Dr. Jaehoon Yu 5 Prefixes, expressions and their meanings deci (d): centi (c): milli (m): micro (μ): nano (n): pico (p): femto (f): atto (a): zepto (z): yocto (y): deca (da): 10 1 hecto (h): 10 2 kilo (k): 10 3 mega (M): 10 6 giga (G): 10 9 tera (T): peta (P): exa (E): zetta (Z): yotta (Y): LargerSmaller

Thursday, June 7, PHYS , Summer 2012 Dr. Jaehoon Yu 6 International Standard Institutes International Bureau of Weights and Measure –Base unit definitions: –Unit Conversions: US National Institute of Standards and Technology (NIST)

Thursday, June 7, PHYS , Summer 2012 Dr. Jaehoon Yu 7 How do we convert quantities from one unit to another? Unit 1 =Unit 2Conversion factor X 1 inch2.54cm 1 inch0.0254m 1 inch2.54x10 -5 km 1 ft30.3cm 1 ft0.303m 1 ft3.03x10 -4 km 1 hr60minutes 1 hr3600seconds And manyMoreHere….

Thursday, June 7, PHYS , Summer 2012 Dr. Jaehoon Yu 8 Examples 1.3 and 1.4 for Unit Conversions Ex 1.3: An apartment has a floor area of 880 square feet (ft 2 ). Express this in square meters (m 2 ). Ex 1.4: Where the posted speed limit is 55 miles per hour (mi/h or mph), what is this speed (a) in meters per second (m/s) and (b) kilometers per hour ( km/h )? (a) What do we need to know? (b)

Thursday, June 7, PHYS , Summer 2012 Dr. Jaehoon Yu 9 Estimates & Order-of-Magnitude Calculations Estimate = Approximation –Useful for rough calculations to determine the necessity of higher precision –Usually done under certain assumptions –Might require modification of assumptions, if higher precision is necessary Order of magnitude estimate: Estimates done to the precision of 10s or exponents of 10s; –Three orders of magnitude: 10 3 =1,000 –Round up for Order of magnitude estimate; 8x10 7 ~ 10 8 –Similar terms: “Ball-park-figures”, “guesstimates”, etc

Thursday, June 7, PHYS , Summer 2012 Dr. Jaehoon Yu 10 Example 1.8 Estimate the radius of the Earth using triangulation as shown in the picture when d=4.4km and h=1.5m. R d=4.4km R+h Pythagorian theorem Solving for R

Thursday, June 7, Dimension and Dimensional Analysis An extremely useful concept in solving physical problems Good to write physical laws in mathematical expressions No matter what units are used the base quantities are the same –Length (distance) is length whether meter or inch is used to express the size: Usually denoted as [L] –The same is true for Mass ([M]) and Time ([T]) –One can say “Dimension of Length, Mass or Time” –Dimensions are treated as algebraic quantities: Can perform two algebraic operations; multiplication or division PHYS , Summer 2012 Dr. Jaehoon Yu

Thursday, June 7, Dimension and Dimensional Analysis cnt’d One can use dimensions only to check the validity of one’s expression: Dimensional analysis –Eg: Speed [v] = [L]/[T]=[L][T -1 ] Distance (L) traveled by a car running at the speed V in time T –L = V*T = [L/T]*[T]=[L] More general expression of dimensional analysis is using exponents: eg. [v]=[L n T m ] =[L][T -1 ] where n = 1 and m = PHYS , Summer 2012 Dr. Jaehoon Yu

Thursday, June 7, Examples Show that the expression [v] = [at] is dimensionally correct Speed: [v] =[L]/[T] Acceleration: [a] =[L]/[T] 2 Thus, [at] = (L/T 2 )xT=LT (-2+1) =LT -1 =[L]/[T]= [v] Dimensionless constant Length Speed rv a Suppose the acceleration a of a circularly moving particle with speed v and radius r is proportional to r n and v m. What are n and m ? PHYS , Summer 2012 Dr. Jaehoon Yu

Thursday, June 7, Some Fundamentals Kinematics: Description of Motion without understanding the cause of the motion Dynamics: Description of motion accompanied with understanding the cause of the motion Vector and Scalar quantities: –Scalar: Physical quantities that require magnitude but no direction Speed, length, mass, height, volume, area, magnitude of a vector quantity, etc –Vector: Physical quantities that require both magnitude and direction Velocity, Acceleration, Force, Momentum It does not make sense to say “I ran with velocity of 10miles/hour.” Objects can be treated as point-like if their sizes are smaller than the scale in the problem –Earth can be treated as a point like object (or a particle)in celestial problems Simplification of the problem (The first step in setting up to solve a problem…) –Any other examples? PHYS , Summer 2012 Dr. Jaehoon Yu

Thursday, June 7, Some More Fundamentals Motions: Can be described as long as the position is known at any given time (or position is expressed as a function of time) –Translation: Linear motion along a line –Rotation: Circular or elliptical motion –Vibration: Oscillation Dimensions –0 dimension: A point –1 dimension: Linear drag of a point, resulting in a line  Motion in one-dimension is a motion on a line –2 dimension: Linear drag of a line resulting in a surface –3 dimension: Perpendicular Linear drag of a surface, resulting in a stereo object PHYS , Summer 2012 Dr. Jaehoon Yu

Thursday, June 7, 2012PHYS , Summer 2012 Dr. Jaehoon Yu 16 Displacement, Velocity and Speed One dimensional displacement is defined as: The average velocity is defined as: The average speed is defined as: Displacement per unit time in the period throughout the motion Displacement is the difference between initial and final potions of the motion and is a vector quantity. How is this different than distance? Unit? m/s Unit? m m/s A vector quantity A scalar quantity A vector quantity

Thursday, June 7, 2012PHYS , Summer 2012 Dr. Jaehoon Yu 17 How much is the elapsed time? What is the displacement?

Thursday, June 7, 2012PHYS , Summer 2012 Dr. Jaehoon Yu 18 Displacement, Velocity and Speed One dimensional displacement is defined as: The average velocity is defined as: The average speed is defined as: Can someone tell me what the difference between speed and velocity is? Displacement per unit time in the period throughout the motion Displacement is the difference between initial and final potions of the motion and is a vector quantity. How is this different than distance? Unit? m/s Unit? m m/s

Thursday, June 7, Difference between Speed and Velocity Let’s take a simple one dimensional translation that has many steps: Let’s call this line as X-axis Let’s have a couple of motions in a total time interval of 20 sec. +10m +15m -15m-5m-10m +5m Total Displacement: Total Distance Traveled: Average Velocity: Average Speed: PHYS , Summer 2012 Dr. Jaehoon Yu

Thursday, June 7, Example 2.1 Displacement: Average Velocity: Average Speed: The position of a runner as a function of time is plotted as moving along the x axis of a coordinate system. During a 3.00-s time interval, the runner’s position changes from x 1 =50.0m to x 2 =30.5 m, as shown in the figure. What was the runner’s average velocity? What was the average speed? PHYS , Summer 2012 Dr. Jaehoon Yu

Thursday, June 7, 2012PHYS , Summer 2012 Dr. Jaehoon Yu 21 Instantaneous Velocity and Speed Can average quantities tell you the detailed story of the whole motion? *Magnitude of Vectors are Expressed in absolute values Instantaneous speed is the size (magnitude) of the velocity vector: Instantaneous velocity is defined as: –What does this mean? Displacement in an infinitesimal time interval Average velocity over a very, very short amount of time NO!!

Thursday, June 7, 2012PHYS , Summer 2012 Dr. Jaehoon Yu 22 Instantaneous Velocity Time Average Velocity Instantaneous Velocity

Thursday, June 7, 2012PHYS , Summer 2012 Dr. Jaehoon Yu 23 Position vs Time Plot time t1t1 t2t2 t3t3 t=0 Position x=0 x1x Running at a constant velocity (go from x=0 to x=x 1 in t1,t1, Displacement is + x1 x1 in t1 t1 time interval) 2.Velocity is 0 (go from x1 x1 to x1 x1 no matter how much time changes) 3.Running at a constant velocity but in the reverse direction as 1. (go from x1 x1 to x=0 in t 3 -t 2 time interval, Displacement is - x1 x1 in t 3 -t 2 time interval) It is helpful to understand motions to draw them on position vs time plots. Does this motion physically make sense?

Thursday, June 7, 2012PHYS , Summer 2012 Dr. Jaehoon Yu 24

Thursday, June 7, 2012PHYS , Summer 2012 Dr. Jaehoon Yu 25 Example 2.3 (a) Determine the displacement of the engine during the interval from t 1 =3.00s to t 2 =5.00s. Displacement is, therefore: A jet engine moves along a track. Its position as a function of time is given by the equation x=At 2 +B where A=2.10m/s 2 and B=2.80m. (b) Determine the average velocity during this time interval.

Thursday, June 7, 2012PHYS , Summer 2012 Dr. Jaehoon Yu 26 Example 2.3 cont’d Calculus formula for derivative The derivative of the engine’s equation of motion is (c) Determine the instantaneous velocity at t=t 2 =5.00s. and The instantaneous velocity at t=5.00s is

Thursday, June 7, 2012PHYS , Summer 2012 Dr. Jaehoon Yu 27 Displacement, Velocity and Speed Displacement Average velocity Average speed Instantaneous velocity Instantaneous speed

Thursday, June 7, 2012PHYS , Summer 2012 Dr. Jaehoon Yu 28 Acceleration In calculus terms: The slope (derivative) of the velocity vector with respect to time or the change of slopes of position as a function of time analogous to analogous to Change of velocity in time (what kind of quantity is this?) Definition of the average acceleration: Definition of the instantaneous acceleration: Vector

Thursday, June 7, 2012PHYS , Summer 2012 Dr. Jaehoon Yu 29 Acceleration vs Time Plot What does this plot tell you? Yes, you are right!! The acceleration of this motion is a constant!!

Thursday, June 7, 2012PHYS , Summer 2012 Dr. Jaehoon Yu 30 Example 2.4 A car accelerates along a straight road from rest to 75km/h in 5.0s. What is the magnitude of its average acceleration?

Thursday, June 7, 2012PHYS , Summer 2012 Dr. Jaehoon Yu 31 Few Confusing Things on Acceleration When an object is moving in a constant velocity ( v=v 0 ), there is no acceleration ( a=0 ) –Is there any acceleration when an object is not moving? When an object is moving faster as time goes on, ( v=v(t) ), acceleration is positive ( a>0 ). –Incorrect, since the object might be moving in negative direction initially When an object is moving slower as time goes on, ( v=v(t) ), acceleration is negative ( a<0 ) –Incorrect, since the object might be moving in negative direction initially In all cases, velocity is positive, unless the direction of the movement changes. –Incorrect, since the object might be moving in negative direction initially Is there acceleration if an object moves in a constant speed but changes direction? The answer is YES!!