Chapter 1: Concepts of Motion

Transcription

1 1.1 Motion diagrams Multiple-exposure photographs with images taken at even time intervals Spacing of images indicative of speed and acceleration speeding up constant speed slowing down SMU PHYS1100.1, Fall 2008, Prof. Clarke 1

2 Assuming equal time intervals between frames, which car is going faster? 1.2 The particle model treat object as though all mass concentrated at a single point Simplifies motion diagram and problem; no longer distracted by unimportant details. SMU PHYS1100.1, Fall 2008, Prof. Clarke 2

3 Which motion diagram best represents a pebble falling through water? a golf ball dropped from a bridge? a man falling on a bungee chord? SMU PHYS1100.1, Fall 2008, Prof. Clarke 3

4 Which motion diagram best represents a pebble falling through water? a golf ball dropped from a bridge? a man falling on a bungee chord? SMU PHYS1100.1, Fall 2008, Prof. Clarke 4

5 1.3 Position and time - position can be represented by a coordinate grid - time can be labeled directly. An arrow drawn from the origin to the object is called a position vector - shown is a vector in its polar representation (length and angle) - Cartesian representation is (4,3) m SMU PHYS1100.1, Fall 2008, Prof. Clarke 5

6 Scalars and vectors: A scalar measures how much (magnitude, no direction)» examples: mass, distance, speed, cost A vector measures magnitude (how much) and direction (which way)» examples: displacement, velocity, acceleration, force The magnitude of a vector is its size or length, and is never negative. It is always positive or zero. Notation: v (with the arrow over it) indicates the full vector v (with no arrow) is the vector magnitude. SMU PHYS1100.1, Fall 2008, Prof. Clarke 6

7 A scalar is a pure number, with or without units: e.g., 3 kg, 14.2 m, 17 A vector is an ordered pair in 2-D, ordered triple in 3-D e.g., v = (3, 4) m (Cartesian); v = (5m, 53 o ) (polar) Direction must be given relative to a coordinate system. e.g., NW, 30 o clockwise from the +y-axis, etc. default is counterclockwise from the +x-axis Vectors may be drawn as an arrow with length proportional to the magnitude, and direction to indicate vector direction. SMU PHYS1100.1, Fall 2008, Prof. Clarke 7 y x

8 Scalars are added only if they have the same units e.g., 3m + 2m = 5m is OK; 2.1kg + 4 is nonsense vector addition vector subtraction SMU PHYS1100.1, Fall 2008, Prof. Clarke 8

9 Which figure shows A 1 + A 2 + A 3? A 1 and A 2 are half the length of A 3 SMU PHYS1100.1, Fall 2008, Prof. Clarke 9

10 Which figure shows A 1 + A 2 + A 3? A 1 and A 2 are half the length of A 3 SMU PHYS1100.1, Fall 2008, Prof. Clarke 10

11 Which figure shows 2A B? SMU PHYS1100.1, Fall 2008, Prof. Clarke 11

12 Which figure shows 2A B? B 2A B 2A SMU PHYS1100.1, Fall 2008, Prof. Clarke 12

13 What are the x- and y-components C x and C y of vector C? a) C x = 3 cm, C y = 1 cm b) C x = 4 cm, C y = 2 cm c) C x = 2 cm, C y = 1 cm d) C x = 3 cm, C y = 1 cm e) C x = 1 cm, C y = 1 cm SMU PHYS1100.1, Fall 2008, Prof. Clarke 13

14 What are the x- and y-components C x and C y of vector C? a) C x = 3 cm, C y = 1 cm b) C x = 4 cm, C y = 2 cm c) C x = 2 cm, C y = 1 cm d) C x = 3 cm, C y = 1 cm e) C x = 1 cm, C y = 1 cm The y-component points up, defined as positive direction. The y-side of the triangle is 2 cm long. The x-component points left, defined as the negative direction. The x-side of the triangle is 4 cm long. ended here, 4/9/08 SMU PHYS1100.1, Fall 2008, Prof. Clarke 14

15 Notes: Chapter 1: Concepts of Motion 1. Course web site: 2. This course was formerly known as PHYS1210, PHY210, PHY205, and PHY PHYS1100 is a calculus-based first-year course in physics, suitable for all physical sciences: physics, astrophysics, chemistry, and engineering. The algebra-based course, PHYS1000, meets at the same time and is suitable for the life sciences. Both count as a science elective. 4. MasteringPhysics Assignment 1 is available until 11:59 pm, Thursday, September 18, SMU PHYS1100.1, Fall 2008, Prof. Clarke 15

16 Displacement, r, is the difference of position vectors. r = r 1 r 0, where r 1 and r 0 are position vectors relative to origin. Even though choice of origin affects position vectors, it does not affect r. SMU PHYS1100.1, Fall 2008, Prof. Clarke 16

17 net displacement = r n = r 1 + r 2 + r 3 total distance = d = r 1 + r 2 + r r 1 = (2, 3) cm r 2 = (2,-2) cm r 3 = (1, 0) cm r n = (5, 1) cm magnitudes: 3 2 r 1 r 2 r 3 r 1 = = 3.61 cm r 2 = 2 2 +(-2) 2 = 2.83 cm r 3 = = 1.00 cm 1 r n r n = = 5.10 cm d = 7.43 cm = r n Distance is not the magnitude of displacement! SMU PHYS1100.1, Fall 2008, Prof. Clarke 17

18 1.4 average speed and average velocity average speed: a scalar defined as the total distance traveled divided by the time taken to travel that distance: s avg = d/ t. average velocity: a vector defined as the net displacement divided by the time taken to undergo that displacement: v avg = r/ t. Since distance is not the magnitude of the net displacement, average speed is not the magnitude of the average velocity. Let r = r 2 r 1, where r 1 is the initial position and r 2 is the position at a time t later. Then v avg = (r 2 r 1 )/ t, and r 2 = r 1 + v avg t SMU PHYS1100.1, Fall 2008, Prof. Clarke 18

19 A particle moves from position 1 to position 2 during the interval t. Which vector shows the particle s average velocity? SMU PHYS1100.1, Fall 2008, Prof. Clarke 19

20 A particle moves from position 1 to position 2 during the interval t. Which vector shows the particle s average velocity? SMU PHYS1100.1, Fall 2008, Prof. Clarke 20

21 On a motion diagram, since the time interval between dots is constant, arrows between dots can equally represent average velocity vectors as they did displacement vectors. Closer dots means slower speeds, further apart dots means faster speeds. SMU PHYS1100.1, Fall 2008, Prof. Clarke 21

22 1.5 Acceleration A motion diagram can indicate changes in the velocity magnitude (top right), direction (bottom right) or both (below). In all cases, the particle is accelerating. As a ball arches through the air, both the magnitude and direction of the velocity change SMU PHYS1100.1, Fall 2008, Prof. Clarke 22

23 average acceleration: a vector defined as the change in velocity of a particle divided by the time it takes for that change to occur: a avg = v/ t Let v = v 2 v 1, where v 1 is the initial velocity and v 2 is the velocity at a time t later. Then a avg = (v 2 v 1 )/ t, and v 2 = v 1 + a avg t On a motion diagram, the (average) acceleration is the difference between two adjacent (average) velocity vectors. It takes three points on a motion diagram to determine one acceleration. SMU PHYS1100.1, Fall 2008, Prof. Clarke 23

24 A particle undergoes acceleration a while moving from point 1 to point 2. Which of the choices shows the velocity vector v 2 as the object moves away from point 2? SMU PHYS1100.1, Fall 2008, Prof. Clarke 24

25 A particle undergoes acceleration a while moving from point 1 to point 2. Which of the choices shows the velocity vector v 2 as the object moves away from point 2? SMU PHYS1100.1, Fall 2008, Prof. Clarke 25

26 speeding up: when the component of a that lies along v points in the same direction as v. slowing down: when the component of a that lies along v points in the opposite direction as v. v v a a Both are accelerating. Physicists don t use the word decelerating. 1.7 Examples of complete motion diagrams A complete motion diagram has: 1. Enough equal-time-spaced dots to illustrate motion 2. The average velocity vectors 3. The average acceleration vectors. SMU PHYS1100.1, Fall 2008, Prof. Clarke 26

27 1. A shot put thrown into the air Turning point INITIAL point in the description of this motion: Object has just left thrower s hand, and has non-zero velocity at the initial point. FINAL point in the description of this motion: Object is just about to hit the ground, but hasn t hit yet (still has non-zero velocity). SMU PHYS1100.1, Fall 2008, Prof. Clarke 27

28 2. A ride on the ferris wheel Length of velocity vectors are all the same ( constant speed ), BUT acceleration vectors are definitely not zero! Find them by subtracting adjacent velocity vectors: all acceleration vectors point to centre of circle! SMU PHYS1100.1, Fall 2008, Prof. Clarke 28

29 1.7 Problem representations A physics problem is represented in a variety of ways: 1. problems are written in a verbal representation; 2. the motion diagram is a physical representation; 3. a drawing is a pictorial representation; 4. there are also mathematical and graphical representations. Each representation has a purpose, and the trick to solving physics problems is to be able to move freely among them. SMU PHYS1100.1, Fall 2008, Prof. Clarke 29

30 Example: throwing a ball into the air. Verbal representation: Consider throwing a ball from your hand directly up into the air. a) What is the velocity of the ball at the turning point? Is the acceleration zero there? b) If the ground is lower than your hand, will the speed of the ball be greater or less than its initial speed just before it hits the ground? Physical representation: on the board Mathematical representation: another day SMU PHYS1100.1, Fall 2008, Prof. Clarke 30

31 1.8 Problem solving strategy 1. model: simplify the situation (e.g., the ball is a particle) 2. visualise: create a pictorial, physical, and/or graphical representation of the problem as needed and appropriate 3. solve: use a mathematical representation (e.g., equations) and solve 4. assess: does your final answer make sense (e.g., ballpark estimates, units, etc.) SMU PHYS1100.1, Fall 2008, Prof. Clarke 31

32 1.9 Significant figures and units Uncertainties are often conveyed simply by the number of significant figures kept in a number. In this class, we will use 3-4 significant figures.» length = 5.12 m OK (means ± about 0.01 m)» length = m not OK (means ± about m)» keep 5 or 6 significant figures in intermediate steps to avoid excessive round-off error.» note that 1.01 has 1/10 the accuracy of 9.94 (think about it!). So, if the first digit is between 1-3, keep 4 significant figures, if the first digit is between 4-9 keep just 3. SMU PHYS1100.1, Fall 2008, Prof. Clarke 32

33 STANDARD UNITS in physics (and in all sciences) are: SI (systeme internationale; metric; mks)» length: m (metres)» mass: kg (kilograms)» time: s (seconds) Need to know the basic metric prefixes: e.g.,» centi (as in cm ) = 10-2» kilo (as in kg ) = 10 3» see TABLES 1.1, 1.2, 1.3 Measurements (and therefore ALL physical quantities) NEED to have the UNITS specified!!!!» length = 5.12 not OK: (inches, feet, metres, miles?) SMU PHYS1100.1, Fall 2008, Prof. Clarke 33