Lecture 4

1. Lecture 4 • Today: Ch. 3 (all) & Ch. 4 (start) • Perform vector algebra • (addition & subtraction) graphically or numerically • Interconvert between Cartesian and Polar coordinates • Begin working with 2D motion • Distinguish position-time graphs from particle trajectory plots • Trajectories • Obtain velocities • Acceleration: Deduce components parallel and perpendicular to the trajectory path Reading Assignment: For Monday read through Chapter 5.3

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. Vectors • In 1 dimension, we can specify direction with a + or - sign. • In 2 or 3 dimensions, we need more than a sign to specify the direction of something: • To illustrate this, consider the position vectorr in 2 dimensions. Example: Where is Boston ? • Choose origin at New York • Choose coordinate system Boston is 212 milesnortheastof New York [ in (r,q) ]OR Boston is 150 milesnorth and 150 mileseast of New York [ in (x,y) ] Boston r New York

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

5. A A = C B C A = B, B = C Vectors • Vectors have both magnitude and a direction • Examples: displacement, velocity, acceleration • The magnitude of A≡ |A| • Usually vectors include units (m, m/s, m/s2) • For vector algebra ( +/ -)a vector has no particular position (Note: the position vector reflects displacement from the origin) • Two vectors are equal if their directions, magnitudes & units match.

6. Scalars • A scalar is an ordinary number. • A magnitude ( can be negative ) without a direction • Often it will have units (i.e. kg) but can be just a number • Usually indicated by a regular letter, no bold face and no arrow on top. Note: Lack of a specific designation can lead to confusion • 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

7. While I conduct my daily run, several quantities describe my condition Which of the following is cannot be a vector ? Exercise Vectors and Scalars • my velocity (3 m/s) • my acceleration downhill (30 m/s2) • my destination (the lab - 100,000 m east) • my mass (150 kg)

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

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

10. U = |U| û û Reference vectors: Unit Vectors • A Unit Vectoris a vector having length 1 and no units • It is used to specify 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 y j x i k z

11. 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

12. 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?

13. 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?

14. tan-1 ( y / x ) Converting Coordinate Systems • 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 cylindrical coordinates (r,q,z), r is the same as the magnitude of the vector in the x-y plane [sqrt(x2 +y2)]

15. Resolving 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 

16. y’ x’ g q q y x Resolving vectors, little g & the inclined plane • g (bold face, vector) can be resolved into its x,yorx’,y’ components • g = - g j • g = - g cos qj’+ g sin qi’ The bigger the tilt the faster the acceleration….. along the incline

17. Dynamics II: Motion along a line but with a twist(2D dimensional motion, magnitude and directions) • Particle motions involve a path or trajectory • Recall instantaneous velocity and acceleration • These are vector expressions reflecting x, y & z motion r = r(t) v = dr / dt a = d2r / dt2

18. 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. • The magnitude of the instantaneous velocity vector is the speed, s. (Knight uses v) s = (vx2 + vy2 + vz )1/2

19. 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). • The average acceleration is a vector quantity directed along ∆v ( a vector! )

20. Instantaneous Acceleration • The instantaneous acceleration is the limit of the average acceleration as ∆v/∆t approaches zero • The instantaneous acceleration is a vector with components parallel (tangential) and/or perpendicular (radial) to the tangent of the path • Changes in a particle’s path may produce an acceleration • The magnitude of the velocity vector may change • The direction of the velocity vector may change (Even if the magnitude remains constant) • Both may change simultaneously (depends: path vs time)

21. = + = 0 = 0 a a a a v v a Generalized motion with non-zero acceleration: need bothpath&time Two possible options: Change in the magnitude of Change in the direction of Animation

22. Kinematics • The position, velocity, and acceleration of a particle in 3-dimensions can be expressed as: r = x i + y j + z k v = vx i + vy j + vz k (i , j , k unit vectors ) a = ax i + ay j + az k • All this complexity is hidden away in r = r(Dt) v = dr / dt a = d2r / dt2

23. 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

24. 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

25. Exercise 1 & 2Trajectories 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)

26. 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

27. B B B C C C B C Exercise 2Trajectories with acceleration • A • B • C • D • None of these From B to C ? A B C D

28. Exercise 3Trajectories with acceleration • A • B • C • D • None of these C C After C ? A B C C C D

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