Physics Fall Mechanics, Thermodynamics, Waves, Fluids. Lecture 6: motion in two and three dimensions III. Slide 6-1

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1 Physics 1501 Fall 2008 Mechanics, Themodynamics, Waves, Fluids Lectue 6: motion in two and thee dimensions III Slide 6-1

2 Recap: elative motion An object moves with velocity v elative to one fame of efeence. That fame moves at V elative to a second efeence fame. Then the velocity of the object elative to the second fame is v = v + V. Example: A jetline flies at 960 km/h elative to the ai, heading nothwad. Thee s a wind blowing eastwad at 190 km/h. In what diection should the plane fly? The vecto diagam identifies the quantities in the equation, and shows that the angle is 11. Slide 6-2

3 Recap: constant acceleation With constant acceleation, the equations fo onedimensional motion apply independently in each diection. The equations take a compact fom in vecto notation. Each equation stands fo two o thee sepaate equations. v = v 0 + at = 0 + v 0 t at 2 Slide 6-3

4 Recap: pojectile motion Motion unde the influence of gavity nea Eath s suface has essentially constant acceleation g whose magnitude is g = 9.8 m/s 2, and whose diection is downwad. Such motion is called pojectile motion. Equations fo pojectile motion, in a coodinate system with y axis vetically upwad: v x = v x0 v y = v y0 gt x = x 0 + v x0 t y = y 0 + v y0 t 1 2 gt 2 Hoizontal and vetical motions ae independent: Slide 6-4

5 Pojectile tajectoies The tajectoy of an object in pojectile motion is a paabola, unless the object has no hoizontal component of motion. Hoizontal motion is unchanged, while vetical motion undegoes downwad acceleation: Equation fo the tajectoy: y = x tanθ 0 g 2v 0 2 cos 2 θ 0 x 2 Slide 6-5

6 Unifom cicula motion When an object moves in a cicula path of adius at constant speed v, its acceleation has magnitude a = v2 The acceleation vecto points towad the cente of the cicle. Since the diection of the acceleation keeps changing, this is not constant acceleation. Slide 6-6

7 question The figue shows velocity vectos fo fou points on a noncicula path. Choose the coect ode, fom smallest to lagest, of the centipetal acceleations at these points given v 1 = v 4 and v 2 = v 3. A. a 1 >a 4 >a 3 >a 2 B. a 2 >a 3 >a 4 >a 1 C. a 3 >a 2 >a 1 >a 4 D. a 2 >a 3 >a 1 >a 4 Slide 6-7

changes both the magnitude and diection of the velocity.

8 Summay In two and thee dimensions, position, velocity, and acceleation become vecto quantities. Velocity is the ate of change of position: Acceleation is the ate of change of velocity: In geneal, acceleation changes both the magnitude and diection of the velocity. v = d dt a = d v dt Pojectile motion esults fom the acceleation of gavity. v In unifom cicula motion, the acceleation has magnitude v 2 / and points towad the cente of the cicula path. Slide 6-8

9 In this lectue you ll lean The concept of foce and its ole in causing changes in motion The fundamental foces of physics Newton s thee laws of motion About the foce of gavity Including the distinction between mass and weight How to apply Newton s laws in onedimensional motion Slide 6-9

10 What causes motion? That s the wong question! The ancient Geek philosophe Aistotle believed that foces pushes and pulls caused motion. The Aistotelian view pevailed fo some 2000 yeas. Galileo and Newton discoveed the coect elation between foce and motion. Foce causes not motion itself but change in motion. The Aistotelian view The Newtonian view Slide 6-10

11 Newton s laws of motion Newton s fist law of motion: A body in unifom motion emains in unifom motion, and a body at est emains at est, unless acted on by a nonzeo net foce. Newton s second law of motion: The ate at which a body s momentum changes is equal to the net foce acting on the body: F net = d p (Newton s 2 dt nd Law) Newton s thid law of motion: If object A exets a foce on object B, then object B exets an oppositely diected foce of equal magnitude on A. Slide 6-11

12 The fist law The fist law is a special case of the second law, when thee s no net foce acting on an object. In that case the object s motion doesn t change. If at est it emains at est. If in motion, it emains in unifom motion. Unifom motion is motion at constant speed in a staight line. Thus the fist law shows that unifom motion is a natual state, equiing no explanation. Slide 6-12

13 question On a hoizontal tabletop is a cuved baie that exets a foce on a ball, guiding its motion in a cicula path as shown. Afte the ball leaves the baie, which of the dashed paths shown does it follow? Slide 6-13

14 The second law The second law tells quantitatively how foce causes changes in an object s quantity of motion. Newton defined quantity of motion, now called momentum, as the poduct of an object s mass and velocity: p = mv Newton s second law equates the ate of change of momentum to the net foce on an object: F = d p dt When mass is constant, Newton s second law becomes ( ) F = d m v dt = m d v dt = m a Slide 6-14

15 question A nonzeo net foce acts on an object. Does that mean the object necessaily moves in the same diection as the net foce? A. Yes B. No Slide 6-15

Centripetal Force. Lecture 11. Chapter 8. Course website:

Centripetal Force. Lecture 11. Chapter 8. Course website: Lectue 11 Chapte 8 Centipetal Foce Couse website: http://faculty.uml.edu/andiy_danylov/teaching/physicsi PHYS.1410 Lectue 11 Danylov Depatment of Physics and Applied Physics Today we ae going to discuss: