Tuesday, June 7, 2011 PHYS 1443-001, Spring 2011 Dr. Jaehoon Yu 1 PHYS 1443 – Section 001 Lecture #2 Tuesday, June 7, 2011 Dr. Jaehoon Yu • Dimensional Analysis • Fundamentals • One Dimensional Motion: Average Velocity; Acceleration; Motion under constant acceleration; Free FallTuesday, June 7, 2011 PHYS 1443-001, Spring 2011 Dr. Jaehoon Yu 2 Announcements • Homework registration and submissions – 18/27 registered but only 1 submitted the answer! – I will then have to approve your enrollment request • So please go ahead and take an action as soon as possible • The roster closes tomorrow, Wednesday! • Quiz tomorrow at the beginning of the class! – Problems will be on Appendices A and B!Tuesday, June 7, 2011 PHYS 1443-001, Spring 2011 Dr. Jaehoon Yu 3 Special Problems for Extra Credit • Derive the quadratic equation for Bx2-Cx+A=0 5 points • Derive the kinematic equation from first principles and the known kinematic equations 10 points • You must show your work in detail to obtain full credit • Due at the start of the class, Thursday, June 9Tuesday, June 7, 2011 4 4 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 1443-001, Spring 2011 Dr. Jaehoon YuTuesday, June 7, 2011 5 5 Dimension and Dimensional Analysis • 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]=[LnTm] =[L][T-1] where n = 1 and m = -1 PHYS 1443-001, Spring 2011 Dr. Jaehoon YuTuesday, June 7, 2011 6 6 Examples • Show that the expression [v] = [at] is dimensionally correct • Speed: [v] =[L]/[T] • Acceleration: [a] =[L]/[T]2 • Thus, [at] = (L/T2)xT=LT(-2+1) =LT-1 =[L]/[T]= [v] Dimensionless constant Length Speed r v a • Suppose the acceleration a of a circularly moving particle with speed v and radius r is proportional to rn and vm. What are n andm? 1PHYS 1443-001, Spring 2011 Dr. Jaehoon YuTuesday, June 7, 2011 7 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, energy, heat, 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 1443-001, Spring 2011 Dr. Jaehoon YuTuesday, June 7, 2011 8 8 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 1443-001, Spring 2011 Dr. Jaehoon YuTuesday, June 7, 2011 PHYS 1443-001, Spring 2011 Dr. Jaehoon Yu 9 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 Unit? m/s A vector quantity A scalar quantity A vector quantityTuesday, June 7, 2011 PHYS 1443-001, Spring 2011 Dr. Jaehoon Yu 10 How much is the elapsed time? What is the displacement? x2Tuesday, June 7, 2011 PHYS 1443-001, Spring 2011 Dr. Jaehoon Yu 11 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 Unit? m/sTuesday, June 7, 2011 12 12 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: ≡ xf − xiPHYS 1443-001, Spring 2011 Dr. Jaehoon YuTuesday, June 7, 2011 13 13 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 x1=50.0m to x2=30.5 m, as shown in the figure. What was the runner’s average velocity? What was the average speed? =50.0 − 30.53.00=19.53.00= 6.50(m / s)PHYS 1443-001, Spring 2011 Dr. Jaehoon YuTuesday, June 7, 2011 PHYS 1443-001, Spring 2011 Dr. Jaehoon Yu 14 Instantaneous
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