2. Physics II, Pg 1 Physics II Today’s Agenda Physics II Today’s Agenda l Newton’s 3 laws. ç How and why do objects move? Dynamics ç Dynamics. l Look at Textbook problems

3. Physics II, Pg 2 Sir Issac Newton

4. Physics II, Pg 3 Dynamics l Issac Newton (1643-1727) published Principia Mathematica in 1687. In this work, he proposed three “laws” of motion: l Law 1: An object subject to no external forces is at rest or moves with a constant velocity if viewed from an inertial reference frame. l FFa Law 2: For any object, F NET =  F = ma l FF l Law 3: Forces occur in pairs: F A,B = - F B,A. See text: 5-1 and 5-2

5. Physics II, Pg 4 Newton’s First Law inertial reference frame l An object subject to no external forces is at rest or moves with a constant velocity if viewed from an inertial reference frame. çIf no forces act, there is no acceleration. l The above statement can be thought of as the definition of inertial reference frames. çAn IRF is a reference frame that is not accelerating (or rotating) with respect to the “fixed stars”. çIf one IRF exists, infinitely many exist since they are related by any arbitrary constant velocity vector! See text: 5-3

6. Physics II, Pg 5 Is Cincinnati a good IRF? l Is Cincinnati accelerating? l YES! çCincinnati is on the Earth. çThe Earth is rotating. l What is the centripetal acceleration of Cincinnati? çT = 1 day = 8.64 x 10 4 sec, çR ~ R E = 6.4 x 10 6 meters. l Plug this in: a U =.034 m/s 2 ( ~ 1/300 g) l Close enough to 0 that we will ignore it. l Cincinnati is a pretty good IRF.

7. Physics II, Pg 6 Newton’s Second Law FFa For any object, F NET =  F = ma. a F çThe acceleration a of an object is proportional to the net force F NET acting on it. çThe constant of proportionality is called “mass”, denoted m. »This is the definition of mass. »The mass of an object is a constant property of that object, and is independent of external influences. l Force has units of [M]x[L/T 2 ] = kg m/s 2 = N (Newton) See text: 5-5

8. Physics II, Pg 7 Newton’s Second Law... l What is a force? çA Force is a push or a pull. çA Force has magnitude & direction (vector). çAdding forces is like adding vectors. FF1FF1 FF2FF2 a FF1FF1 FF2FF2 a F F NET Fa F NET = ma See text: 5-5 and 5-7

9. Physics II, Pg 8 Newton’s Second Law... Fa l Components of F = ma : F X = ma X F Y = ma Y F Z = ma Z l Suppose we know m and F X, we can solve for a X and apply the things we learned about kinematics over the last few weeks: See text: 5-5, 5-6, and 5-7

10. Physics II, Pg 9 Example: Pushing a Box on Ice. i l A skater is pushing a heavy box (mass m = 100 kg) across a sheet of ice (horizontal & frictionless). He applies a force of 50N in the i direction. If the box starts at rest, what is it’s speed v after being pushed a distance d=10m ? F v = 0 m a i See text: 5-5

11. Physics II, Pg 10 Example: Pushing a Box on Ice. i l A skater is pushing a heavy box (mass m = 100 kg) across a sheet of ice (horizontal & frictionless). He applies a force of 50N in the i direction. If the box starts at rest, what is it’s speed v after being pushed a distance d=10m ? d F v m a i See text: 5-5

12. Physics II, Pg 11 Example: Pushing a Box on Ice... l Start with F = ma. ça = F / m. çRecall that v 2 2 - v 1 2 = 2a(x 2 - x 1 )(lecture 1) çSo v 2 = 2Fd / m d F v m a i See text: 5-5

13. Physics II, Pg 12 Example: Pushing a Box on Ice... l Plug in F = 50N, d = 10m, m = 100kg: çFind v = 3.2 m/s. d F v m a i See text: 5-5

14. Physics II, Pg 13 Forces Units of force (mks): [F] = [m][a] = kg m s  2 = N (Newton) l We will consider two kinds of forces: çContact force: »This is the most familiar kind. n I push on the desk. n The ground pushes on the chair... çAction at a distance (a bit mysterious): »Gravity »Electromagnetic, strong & weak nuclear forces. See text: 5-4

15. Physics II, Pg 14 Contact forces: l Objects in contact exert forces. F l Convention: F a,b means “the force acting on a due to b”. F l So F head,thumb means “the force on the head due to the thumb”. F F head,thumb See text: 5-4

16. Physics II, Pg 15 Gravity... l Near the earth’s surface... Fa l But we have just learned that: F g = ma çThis must mean that g is the “acceleration due to gravity” that we already know! Fgg l So, the force on a mass m due to gravity near the earth’s surface is F g = mg where g is 9.8m/s 2 “down”. and See text: 9-2

17. Physics II, Pg 16 Example gravity problem: l What is the force of gravity exerted by the earth on a typical physics student? çTypical student mass m = 55kg çg = 9.8 m/s 2. çF g = mg = (55 kg)x(9.8 m/s 2 ) çF g = 539 N l The force that gravity exerts on any object is called its Weight FFgFFg See text: 5-6 See text example Mass and Weight.

18. Physics II, Pg 17 Newtons Third Law: FF l Forces occur in pairs: F A,B = - F B,A. çFor every “action” there is an equal and opposite “re- action”. l In the case of gravity: R 12 m1m1 m2m2 F F 12 F F 21 See text: 5-8

19. Physics II, Pg 18 Newtons Third Law... FF l F A,B = - F B,A. is true for contact forces as well: F F m,w F F w,m F F m,f F F f,m See text: 5-8

20. Physics II, Pg 19 Example of Bad Thinking FFFa l Since F m,b = -F b,m why isn’t F net = 0, and a = 0 ? a ?? F F m,b F F b,m ice See text: 5-8

21. Physics II, Pg 20 Example of Good Thinking only the box l Consider only the box as the system! çFaF çF on box = ma box = F b,m çFree Body Diagram (next time). a box F F m,b F F b,m ice See text: 5-8

22. Physics II, Pg 21 The Free Body Diagram Fa l Newtons 2nd says that for an object F = ma. for an object. l Key phrase here is for an object. Fa l So before we can apply F = ma to any given object we isolate the forces acting on this object: See text: 5-8, 6-1

23. Physics II, Pg 22 The Free Body Diagram... l Consider a plank leaning against a wall. çWhat are the forces acting on the plank ? çP = plank çF = floor çW = wall çE = earth F F PW F F WP F F PF F F PE F F FP F F EP See text: 5-8, 6-1

24. Physics II, Pg 23 The Free Body Diagram... l Consider the previous case çWhat are the forces acting on the plank ? Isolate the plank from the rest of the world. F F PW F F WP F F PF F F PE F F FP F F EP See text: 5-8, 6-1

25. Physics II, Pg 24 The Free Body Diagram... l The forces acting on the plank should reveal themselves... F F PW F F PF F F PE See text: 5-8, 6-1

26. Physics II, Pg 25 Aside... l In this example the plank is not moving... çIt is certainly not accelerating! Fa F çSo F NET = ma becomes F NET = 0 çThis is the basic idea behind statics, which we will discuss in a few weeks. F F PW F F PF F F PE FFF F PW + F PF + F PE = 0 See text: 6-1

27. Physics II, Pg 26 Example l Example dynamics problem: i A box of mass m = 2kg slides on a horizontal frictionless floor. A force F x = 10N pushes on it in the i direction. What is the acceleration of the box? Fi F = F x i a a = ? m j i See text: 6-1

28. Physics II, Pg 27 Example... l Draw a picture showing all of the forces F F F BF F F FB F F BE F F EB j i See text: 6-1

29. Physics II, Pg 28 Example... l Draw a picture showing all of the forces. l Isolate the forces acting on the block. F F F BF F F FB Fg F BE = mg F F EB j i See text: 6-1

30. Physics II, Pg 29 Example... l Draw a picture showing all of the forces. l Isolate the forces acting on the block. l Draw a free body diagram. F F F BF gmggmg j i See text: 6-1

31. Physics II, Pg 30 Example... l Draw a picture showing all of the forces. l Isolate the forces acting on the block. l Draw a free body diagram. l Solve Newtons equations for each component. ç F X = ma X ç F BF - mg = ma Y F F F BF gmggmg j i See text: 6-1 See strategy: Solving Newton’s Law Problems,

32. Physics II, Pg 31 Example... l F X = ma X ç So a X = F X / m = (10 N)/(2 kg) = 5 m/s 2. l F BF - mg = ma Y ç But a Y = 0 ç So F BF = mg. Normal Force l The vertical component of the force of the floor on the object (F BF ) is often called the Normal Force (N). l Since a Y = 0, N = mg in this case. FXFX N mg j i See text: 6-1

33. Physics II, Pg 32 Example Recap FXFX N = mg mg a X = F X / m j i See text: 6-1

34. Physics II, Pg 33 Tools: Ropes & Strings l Can be used to pull from a distance. l Tension l Tension (T) at a certain position in a rope is the magnitude of the force acting across a cross-section of the rope at that position. çThe force you would feel if you cut the rope and grabbed the ends. çAn action-reaction pair. cut T T T See text: 6-1

35. Physics II, Pg 34 Tools: Ropes & Strings... l An ideal (massless) rope has constant tension along the rope. l If a rope has mass, the tension can vary along the rope ç For example, a heavy rope hanging from the ceiling... l We will deal mostly with ideal massless ropes. T = T g T = 0 TT

36. Physics II, Pg 35 Tools: Ropes & Strings... l The direction of the force provided by a rope is along the direction of the rope: mg T m Since a y = 0,(not moving) T = mg See text: 6-1

37. Physics II, Pg 36 Scales: l Springs can be calibrated to tell us the applied force. ç We can calibrate scales to read Newtons, or... çFishing scales usually read weight in kg or lbs. 0 2 4 6 8 See text: 5-9

38. Physics II, Pg 37 Tools: Pegs & Pulleys l Used to change the direction of forces. çAn ideal massless pulley or ideal smooth peg will change the direction of an applied force without altering the magnitude: FF1FF1 ideal peg or pulley FF2FF2 FF | F 1 | = | F 2 |

39. Physics II, Pg 38 Tools: Pegs & Pulleys l Used to change the direction of forces. çAn ideal massless pulley or ideal smooth peg will change the direction of an applied force without altering the magnitude: mg T m T = mg F W,S = mg

40. Physics II, Pg 39 Recap of today’s lecture l Newton’s 3 laws: Law 1: An object subject to no external forces is at rest or moves with a constant velocity if viewed from an inertial reference frame. FFa Law 2: For any object, F NET =  F = ma FF Law 3: Forces occur in pairs: F A,B = - F B,A.