1 / 27

Lecture 4

Lecture 4. Today: Ch. 3 (all) & Ch. 4 (start) Perform vector algebra (addition and subtraction) Interconvert between Cartesian and Polar coordinates Work with 2D motion Deconstruct motion into x & y or parallel & perpendicular Obtain velocities Obtain accelerations

Download Presentation

Lecture 4

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Lecture 4 • Today: Ch. 3 (all) & Ch. 4 (start) • Perform vector algebra (addition and subtraction) • Interconvert between Cartesian and Polar coordinates • Work with 2D motion • Deconstruct motion into x & y or parallel & perpendicular • Obtain velocities • Obtain accelerations Deduce components parallel and perpendicular to the trajectory path

  2. Coordinate Systems and vectors • In 1 dimension, only 1 kind of system, • Linear Coordinates (x) +/- • In 2 dimensions there are two commonly used systems, • Cartesian Coordinates (x,y) • Circular Coordinates (r,q) • In 3 dimensions there are three commonly used systems, • Cartesian Coordinates (x,y,z) • Cylindrical Coordinates (r,q,z) • Spherical Coordinates (r,q,f)

  3. A A Vectors look like... • There are two common ways of indicating that something is a vector quantity: • Boldface notation: A • “Arrow” notation: Aor

  4. A B C Vectors act like… • Vectors have both magnitude and a direction • Vectors: position, displacement, velocity, acceleration • Magnitude of a vector A≡ |A| • For vector addition or subtraction we can shift vector position at will (NO ROTATION) • Two vectors are equal if their directions, magnitudes & units match. A = C A ≠B, B ≠ C

  5. Scalars • A scalar is an ordinary number. • A magnitude ( + or - ), but no direction • May have units (e.g. kg) but can be just a number • No bold face and no arrow on top. • The product of a vector and a scalar is another vector in the same “direction” but with modified magnitude B A = -0.75 B A

  6. Vectors and 2D vector addition • The sum of two vectors is another vector. A = B + C B B A C C

  7. B C 2D Vector subtraction • Vector subtraction can be defined in terms of addition. = B + (-1)C B - C

  8. B - C -C B Different direction and magnitude ! B+C 2D Vector subtraction • Vector subtraction can be defined in terms of addition. = B + (-1)C B - C B -C

  9. U = |U| û û Unit Vectors • A Unit Vectorpoints : a length 1 and no units • Gives a direction. • Unit vector u points in the direction of U • Often denoted with a “hat”: u = û • Useful examples are the cartesian unit vectors [i, j, k] or • Point in the direction of the x, yand z axes. R = rx i + ry j + rz k or R = x i + y j + z k y j x i k z

  10. By C B Bx A Ay Ax Vector addition using components: • Consider, in 2D, C =A + B. (a) C = (Ax i + Ayj ) + (Bxi + Byj ) = (Ax + Bx )i + (Ay + By ) (b)C = (Cx i+ Cy j ) • Comparing components of (a) and (b): • Cx = Ax + Bx • Cy = Ay + By • |C| =[(Cx)2+ (Cy)2 ]1/2

  11. ExampleVector Addition • {3,-4,2} • {4,-2,5} • {5,-2,4} • None of the above • Vector A = {0,2,1} • Vector B = {3,0,2} • Vector C = {1,-4,2} What is the resultant vector, D, from adding A+B+C?

  12. tan-1 ( y / x ) Converting Coordinate Systems (Decomposing vectors) • In polar coordinates the vector R= (r,q) • In Cartesian the vectorR= (rx,ry) = (x,y) • We can convert between the two as follows: y (x,y) r ry  rx x • In 3D

  13. Decomposing vectors into componentsA mass on a frictionless inclined plane • A block of mass m slides down a frictionless ramp that makes angle  with respect to horizontal. What is its acceleration a ? m a 

  14. Decomposing vectors into componentsA mass on a frictionless inclined plane • A block of mass m slides down a frictionless ramp that makes angle  with respect to horizontal. What is its acceleration a ? m g sin  g = - g j -g cos  

  15. Motion in 2 or 3 dimensions • Position • Displacement • Velocity (avg.) • Acceleration (avg.)

  16. v Instantaneous Velocity • But how we think about requires knowledge of the path. • The direction of the instantaneous velocity is alonga line that is tangent to the path of the particle’s direction of motion.

  17. Average Acceleration • The average acceleration of particle motion reflects changes in the instantaneous velocity vector (divided by the time interval during which that change occurs). • Instantaneous acceleration

  18. = + a a a Tangential ║ & radial acceleration Acceleration outcomes: If parallel changes in the magnitude of (speeding up/ slowing down) Perpendicular changes in the direction of (turn left or right)

  19. Kinematics • In 2-dim. position, velocity, and acceleration of a particle: r = x i + y j v = vx i + vy j(i , j unit vectors ) a = ax i + ay j • All this complexity is hidden away in r = r(Dt) v = dr / dt a = d2r / dt2

  20. xvst x 4 0 t Special Case Throwing an object with x along the horizontal and y along the vertical. x and y motion both coexist and t is common to both Let g act in the –y direction, v0x= v0 and v0y= 0 xvsy yvst t = 0 y 4 y x 4 t 0

  21. Another trajectory Can you identify the dynamics in this picture? How many distinct regimes are there? Are vx or vy = 0 ? Is vx >,< or = vy ? xvsy t = 0 y t =10 x

  22. Another trajectory Can you identify the dynamics in this picture? How many distinct regimes are there? 0 < t < 3 3 < t < 7 7 < t < 10 • I. vx = constant = v0 ; vy = 0 • II. vx = vy = v0 • III. vx = 0 ; vy = constant < v0 xvsy t = 0 What can you say about the acceleration? y t =10 x

  23. Exercises 1 & 2 Trajectories with acceleration • A rocket is drifting sideways (from left to right) in deep space, with its engine off, from A to B. It is not near any stars or planets or other outside forces. • Its “constant thrust” engine (i.e., acceleration is constant) is fired at point B and left on for 2 seconds in which time the rocket travels from point B to some point C • Sketch the shape of the path from B to C. • At point C the engine is turned off. • Sketch the shape of the path after point C (Note: a = 0)

  24. B B B C C C B C Exercise 1Trajectories with acceleration • A • B • C • D • None of these From B to C ? A B C D

  25. Exercise 2Trajectories with acceleration • A • B • C • D • None of these C C After C ? A B C C C D

  26. Exercise 2Trajectories with acceleration • A • B • C • D • None of these C C After C ? A B C C C D

  27. Lecture 4 Assignment: Read all of Chapter 4, Ch. 5.1-5.3

More Related