N1 Newton s Laws and N2 - Vector Calculus. General Physics Volume N 1

Transcription

1 N1 Newton s Laws and N2 - Vector Calculus General Physics Volume N 1

2 Newton s First Law Total momentum of an isolated system is conserved. Mv CM = p r 1 + p r p r N = p r tot In the absence of external interactions, an object s or system s center of mass moves at a constant velocity. General Physics Volume N 2

3 Newton s Third Law A tiny impulse [ p ] out of A is the same as - [ dp r ] flowing into A r [ p ] # [ p ]& F B = dr d r = "" dr $ = " r ' When objects A and B interact, the force of the interaction exerts on A is equal in magnitude and opposite in direction to the force that it exerts on B. F A General Physics Volume N 3

4 Two Minute Problem - N1T.3 A large car drags a small trailer in such a way that their common speed increases rapidly. Which tugs harder on the other? A. The car tugs harder on the trailer than vice versa. B. The trailer tugs harder on the car than vice versa. C. Both tug equally on each other D. The trailer exerts no force on the car at all E. There is no way to tell General Physics Volume N 4

5 Two Minute Problem - N1T.4 A parent pushes a small child on a swing so that the child moves rapidly away while the parent remains at rest. How does the magnitude of the force that the child exerts on the parent compare to the magnitude of the force that the parent exerts on the child? A. The force on the child is larger B. The force on the parent is larger C. These forces have equal magnitudes D. The child exerts zero force on the parent E. There is no way to tell General Physics Volume N 5

6 Two Minute Problem - N1T.6 Arrows representing four forces having equal magnitudes are shown below. What combinations of these forces, acting together on the same object, will allow that object to move with a constant velocity? r F 1 r F 2 r F 3 r F 4 r F 5 A. F2 and F3 B. F3 and F4 C. V1, F2, and F4 D. F1, F3, and F4 E. A and D F. None of the above General Physics Volume N 6

7 Newton s Second Law The next external force on an object causes its mass to accelerate r F net,ext = dr p tot = M dr v CM " M r a CM One of the most important equations in physics! General Physics Volume N 7

8 Classification of Forces General Physics Volume N 8

9 Free-Body Diagrams Draw interaction diagram/conceptual diagram Draw object alone For each force, draw a dot where the force is applied and an arrow representing magnitude and direction Arrows only for forces Every force arrow reflects an interaction General Physics Volume N 9

10 Free-body diagram - book on a table Interaction/conceptual diagram Draw object Draw dot for each force From dot draw direction of each force General Physics Volume N 10

11 Vector Calculus Why? In order to better understand velocity and acceleration Newton had to invent calculus to explain his theory! Review in appendix NA and NB at end of Volume N General Physics Volume N 11

12 Time Derivative of a Vector The time derivative is defined as df " lim #t $ 0 ) f t) f t + #t #t df "f The difference between and is that "t The first expression is at an instant in time The second expression is for a short time interval General Physics Volume N 12

13 Derivative of a Vector The components of the time derivative vector are the ordinary time derivatives of the components of that vector d r q " # $ dq x dq y dq z & ' General Physics Volume N 13

14 Definition of Velocity The velocity vector is the derivative of the displacement vector r v " dr r " small displacement short time interval # r v " dr r " $ dx dy dz & ' v = mag r v ) = v x 2 + v y 2 + v z 2 = " $ # dx ' & 2 " + dy $ ' # & 2 " + dz $ ' # & 2 General Physics Volume N 14

15 Velocity - Instantaneous and Average General Physics Volume N 15

16 Definition of Acceleration An object s acceleration at an instant is a vector that expresses how rapidly and in what direction its velocity vector is changing at that instant r a " dr v " dv x dv y dv z General Physics Volume N 16 # $ & '

17 Motion Diagrams General Physics Volume N 17

18 Motion Map - Acceleration General Physics Volume N 18

19 Two Minute Problem - N2T.4 An object can have a constant speed and still be accelerating, true T) or false F)? General Physics Volume N 19

20 Motion Diagram Example General Physics Volume N 20

21 Two Minute Problem - N2T.5 An object s acceleration vector always points in the direction that it is moving, true T) or false F)? General Physics Volume N 21

22 Motion Map - 2D General Physics Volume N 22

23 Uniform Circular Motion The magnitude of the acceration of an object in uniform circular motion a " mag r a ) = mag v r ) 2 Depends on v and R Points toward center of circle R = v 2 R General Physics Volume N 23

24 General Physics Volume N 24

25 Group Problems Free body diagrams Object slides at constant speed without friction. Object slows due to kinetic friction. Object slides without friction. The object is pulled by a force parallel to the surface. The object is pushed by a force applied downward at an angle. The ball is rising in a parabolic trajectory. General Physics Volume N 25

26 Group Activity - Make a Motion Map Throw the ball vertically upward Not too high Using LoggerPro, capture the throw On paper, draw a motion map How does the velocity change? Draw velocity vectors Draw acceleration vectors General Physics Volume N 26

27 Group Problems N1B.3 N1B.6 N1S.2 N2B.1 N2B.6 N2B.8 General Physics Volume N 27