Lecture 4 Interconvert between Cartesian and Polar coordinates 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 1

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)

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

A = C A ≠B, 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 B C A = C A ≠B, B ≠ C

A scalar is an ordinary number. 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

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

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

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

Unit Vectors û A Unit Vector points : 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 U = |U| û û Useful examples are the cartesian unit vectors [ i, j, k ] or Point in the direction of the x, y and z axes. R = rx i + ry j + rz k or R = x i + y j + z k y j x i k z

Vector addition using components: Consider, in 2D, C = A + B. (a) C = (Ax i + Ay j ) + (Bx i + By j ) = (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 C Bx A By B Ax Ay

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

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

Decomposing vectors into components A 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 

Decomposing vectors into components A 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 

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

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

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

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)

Kinematics All this complexity is hidden away in 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

x vs y x vs t y vs t Special Case x y y t x t 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 x vs y t x 4 x vs t y vs t t = 0 y 4 y 4 t x

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

x vs y Another trajectory y x 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 x vs y t = 0 What can you say about the acceleration? y t =10 x

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. after point C (Note: a = 0)

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

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

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

Assignment: Read all of Chapter 4, Ch. 5.1-5.3 Lecture 4 Assignment: Read all of Chapter 4, Ch. 5.1-5.3 1