Physics 207, Lecture 4, Sept. 15

Transcription

1 Phsics 07, Lecture 4, Sept. 15 Goals for hapts.. 3 & 4 Perform vector algebra (addition & subtraction) graphicall or b, & z components Interconvert between artesian and Polar coordinates Distinguish position-time time graphs from particle trajector plots Obtain particle velocities and accelerations from trajector plots Determine both magnitude and acceleration parallel and perpendicular to the trajector path Solve problems with multiple accelerations in -dimensions (including linear, projectile and circular motion) Discern different reference frames and understand how the relate to particle motion in stationar and moving frames Phsics 07: Lecture 4, Pg 1 Phsics 07, Lecture 4, Sept. 15 Assignment: Read hapter 5 (sections carefull) MP Problem Set due Wednesda (should have started) MP Problem Set 3, hapters 4 and 5 (available toda) Phsics 07: Lecture 4, Pg Page 1

2 Vector addition The sum of two vectors is another vector. A + A Phsics 07: Lecture 4, Pg 3 Vector subtraction Vector subtraction can be defined in terms of addition. - + (-1) - - A + A Different direction and magnitude! Phsics 07: Lecture 4, Pg 4 Page

3 Unit Vectors A Unit Vector is a vector having length 1 and no units It is used to specif a direction. Unit vector u points in the direction of U Often denoted with a hat : u û û U U û Useful eamples are the cartesian unit vectors [ i, j, k ] Point in the direction of the, and z aes. R r i + r j + r z k z k j i Phsics 07: Lecture 4, Pg 5 Vector addition using components: onsider, in D, A +. (a) (A i + A j ) + ( i + j ) (A + )i + (A + ) (b) ( i + j ) omparing components of (a) and (b): A + A + [ ( ) + ( ) ] 1/ A A A Phsics 07: Lecture 4, Pg 6 Page 3

4 Vector A {0,,1} Vector {3,0,} Vector {1,-4,} Eercise 1 Vector Addition What is the resultant vector, D, from adding A++? A. {3,-4,}. {4,-,5}. {5,-,4} D. None of the above Phsics 07: Lecture 4, Pg 7 onverting oordinate Sstems In polar coordinates the vector R (r,θ) In artesian the vector R (r,r ) (,) We can convert between the two as follows: r r r cos θ r cos θ (,) R î + ĵ r + r r θ θ tan -1 ( / ) In 3D clindrical coordinates (r,θ,z), r is the same as the magnitude of the vector in the - plane [sqrt( + )] r Phsics 07: Lecture 4, Pg 8 Page 4

5 Eercise: Frictionless inclined plane A block of mass m slides down a frictionless ramp that makes angle θ with respect to horizontal. What is its acceleration a? m a θ Phsics 07: Lecture 4, Pg 9 Resolving vectors, little g & the inclined plane g θ g (bold face, vector) can be resolved into its, or, components g - g j g - g cos θ j + g sin θ i θ The bigger the tilt the faster the acceleration.. along the incline Phsics 07: Lecture 4, Pg 10 Page 5

6 Dnamics II: Motion along a line but with a twist (D dimensional motion, magnitude and directions) Particle motions involve a path or trajector Recall instantaneous velocit and acceleration These are vector epressions reflecting, & z motion r r(t) v dr / dt a d r / dt Phsics 07: Lecture 4, Pg 11 Instantaneous Velocit ut how we think about requires knowledge of the path. The direction of the instantaneous velocit is along a line that is tangent to the path of the particle s direction of motion. The magnitude of the instantaneous velocit vector is the speed, s. (Knight uses v) s (v + v + v z ) 1/ v Phsics 07: Lecture 4, Pg 1 Page 6

7 Average Acceleration The average acceleration of particle motion reflects changes in the instantaneous velocit vector (divided b the time interval during which that change occurs). The average acceleration is a vector quantit directed along v ( a vector! ) a Phsics 07: Lecture 4, Pg 13 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 hanges in a particle s path ma produce an acceleration The magnitude of the velocit vector ma change The direction of the velocit vector ma change (Even if the magnitude remains constant) oth ma change simultaneousl (depends: path vs time) Phsics 07: Lecture 4, Pg 14 Page 7

8 Motion along a path ( displacement, velocit, acceleration ) or 3D Kinematics : vector equations: r r( t) v dr / dt a d r / dt r r r v + v 1 v v r v r 1 - v r 1 v path Velocit : v av r / t v dr / dt Acceleration : a av v / t a dv / dt Phsics 07: Lecture 4, Pg 15 Generalized motion with non-zero acceleration: r a t r a r r a r a v r a r Two possible options: hange in the magnitude of v hange in the direction of Animation r r a + 0 with a a r at need both path & time a a + v a a 0 a 0 Q1: What is the time sequence in this particle s motion? Q: Is the particle speeding up or slowing down? Phsics 07: Lecture 4, Pg 16 Page 8

9 Eamples of motion: hemotais An eample.. Q1: How do single cell animals locate their food? Possible mechanism: Tumble and run paradigm. Q: How does a fl find its meal? Possible mechanism: If ou smell it fl into the wind, if ou don t fl across the wind. Phsics 07: Lecture 4, Pg 17 Kinematics The position, velocit, and acceleration of a particle in with, 0 3-dimensions can be epressed as: r i + j + z k if d v dt d dt v v i + v j + v z k (i, j, k unit vectors ) a a i + a j + a z k ( t) ( t) z z( t) a constant accel., d dz v v z dt dt d d z a a z dt dt e.g. ( t) All this compleit is hidden awa in r r( t) v dr / dt a d r / dt 0 + v 1 t + a t Phsics 07: Lecture 4, Pg 18 Page 9

10 Special ase Throwing an object with along the horizontal and along the vertical. and motion both coeist and t is common to both Let g act in the direction, v 0 v 0 and v 0 0 vs t vs t t 0 vs t 0 4 t Phsics 07: Lecture 4, Pg 19 Another trajector an ou identif the dnamics in this picture? How man distinct regimes are there? Are v or v 0? Is v >,< or v? t 0 vs t 10 Phsics 07: Lecture 4, Pg 0 Page 10

11 t 0 vs Another trajector an ou identif the dnamics in this picture? How man distinct regimes are there? 0 < t < 3 3 < t < 7 7 < t < 10 I. v constant v 0 ; v 0 II. v v v 0 III. v 0 ; v constant < v 0 What can ou sa about the acceleration? t 10 Phsics 07: Lecture 4, Pg 1 Eercise & 3 Trajectories with acceleration A rocket is drifting sidewas (from left to right) in deep space, with its engine off, from A to. It is not near an stars or planets or other outside forces. Its constant thrust engine (i.e., acceleration is constant) is fired at point and left on for seconds in which time the rocket travels from point to some point Sketch the shape of the path from to. At point the engine is turned off. Sketch the shape of the path after point Phsics 07: Lecture 4, Pg Page 11

12 Eercise Trajectories with acceleration From to? A. A.. D. D E. None of these A D Phsics 07: Lecture 4, Pg 3 Eercise 3 Trajectories with acceleration After? A. A.. D. D E. None of these A D Phsics 07: Lecture 4, Pg 4 Page 1

13 Trajector with constant acceleration along the vertical How does the trajector appear to different observers? What if the observer is moving with the same velocit (i.e. running in parallel)? vs t t 0 vs t Phsics 07: Lecture 4, Pg 5 Trajector with constant acceleration along the vertical This observer will onl see the motion t 0 vs t t 0 vs 4 4 In an inertial reference frame all see the same acceleration Phsics 07: Lecture 4, Pg 6 Page 13

14 Trajector with constant acceleration along the vertical What do the velocit and acceleration vectors look like? Velocit vector is alwas tangent to the curve! Acceleration ma or ma not be! t 0 vs 4 Phsics 07: Lecture 4, Pg 7 Home Eercise The Pendulum 1 θ 30 Which statement best describes the motion of the pendulum bob at the instant of time drawn? i. when the bob is at the top of its swing. ii. which quantities are non-zero? A) v r 0 a r 0 v T 0 a T 0 ) v r 0 a r 0 v T 0 a T 0 ) v r 0 a r 0 v T 0 a T 0 Phsics 07: Lecture 4, Pg 8 Page 14

15 Home Eercise The Pendulum Solution mg T a a T O θ 30 NOT uniform circular motion : If circular motion then a r not zero, If speed is increasing so a T not zero However, at the top of the swing the bob temporaril comes to rest, so v 0 and the net tangential force is mg sin θ ) v r 0 a r 0 v T 0 a T 0 Everwhere else the bob has a nonzero velocit and so then (ecept at the bottom of the swing) v r 0 a r 0 v T 0 a T 0 Phsics 07: Lecture 4, Pg 9 Phsics 07, Lecture 4, Sept. 15 Assignment: Read hapter 5 (sections carefull) MP Problem Set due Wednesda (should have started) MP Problem Set 3, hapters 4 and 5 (available toda) Phsics 07: Lecture 4, Pg 30 Page 15