The Laws of Motion: Understanding Kinematics, Dynamics, and Forces

1. Chapter 4 The Laws of Motion Physics 350

2. The Laws of Motion • Kinematics • Math of HOW things move • Position, velocity, acceleration • Dynamics • WHY do things move? • What causes a body to accelerate? • Forces => Acceleration • The properties of force and the relationships between force and acceleration are given by Newton’s Three Laws of Motion

3. The Laws of Motion • The First Law describes the natural state of motion of a body on which no forces are acting. The other two laws describe the behavior of a body under the influence of forces. • Early 1600’s, theories of object’s tendency to be at rest

4. Laws of Motion • Galileo Galilei • Developed first correct ideas of motion • Gravity and constant acceleration • Forces acting on bodies • Sir Isaac Newton • 1687 Principia Mathematica • Laws of Motion • Law of Universal Gravitation • Invented calculus to further describe speed, acceleration

5. Forces • Force • A central concept in all of physics • A vector quantity • Magnitude • Direction • Force is used to describe push or a pull • Forces on objects • springs • rubber bands • ropes • Cables

6. Forces (cont.) • Force • Bouyant Forces • liquids • Friction • Surfaces • All examples above are known as “contact forces”

7. Newton’s First Law • Newton’s First Law “An object moves with a velocity that is constant in magnitude and direction, unless acted on by a nonzero net force.” • The net force on an object is defined as the vector sum of all external forces exerted on the object. • Often called the Law of inertia • Tendency of object in motion to stay in motion

8. ConcepTest 4.2Cart on Track I 1) slowly come to a stop 2) continue with constant acceleration 3) continue with decreasing acceleration 4) continue with constant velocity 5) immediately come to a stop Consider a cart on a horizontal frictionless table. Once the cart has been given a push and released, what will happen to the cart?

9. ConcepTest 4.2Cart on Track I 1) slowly come to a stop 2) continue with constant acceleration 3) continue with decreasing acceleration 4) continue with constant velocity 5) immediately come to a stop Consider a cart on a horizontal frictionless table. Once the cart has been given a push and released, what will happen to the cart? After the cart is released, there is no longer a force in the x-direction. This does not mean that the cart stops moving!! It simply means that the cart will continuemoving with the same velocity it had at the moment of release. The initial push got the cart moving, but that force is not needed to keep the cart in motion.

10. Newton’s Second Law • Newton’s Second Law “The acceleration a of an object is directly proportional to the net force acting on it and inversely proportional to its mass.” • What it means: a = ΣF / m or conversely, ΣF = ma

11. Newton’s Second Law ΣF = ma • A LAW of nature! • Precise definition of FORCE • a and F are in the same direction • after F is completely summed

12. Newton’s Second Law (cont.) ΣF = ma • ΣF • ΣFx= max • ΣFy= may • ΣFz= maz • No net force, means acceleration is zero • Velocity is constant

13. Newton’s Second Law (cont.) • Units • Mass • Kilograms kg • Acceleration • m/s2 • Force • Newtons 1N = 1 kg x m/s2 (m x a) • Pounds 1N = 0.225lb • Pound is defined as F = ma = slug x ft/s2

14. Newton’s Second Law Definition of Mass (physics) A measure of resistance a body offers to changes in its velocity (acceleration) • Standard is kilogram • Masses can be compared by balances • Mass vs. Weight

15. Newton’s 2nd Lawproves that different masses accelerate to the earth at the same rate, but with different forces. • We know that objects with different masses accelerate to the ground at the same rate. • However, because of the 2nd Law we know that they don’t hit the ground with the same force. F = ma 98 N = 10 kg x 9.8 m/s/s F = ma 9.8 N = 1 kg x 9.8 m/s/s

16. Newton’s Second Law (cont.) • Example A mass of 0.2kg slides along the table with a velocity of v = 2.8m/s. It stops in 1.0 m. What force is acting on the mass (neglect friction)?

17. Newton’s Third Law • Newton’s Third Law • “If object 1 and object 2 interact, the force F12 exerted by object 1 on object 2 is equal in magnitude but opposite to the force F21 exerted by object 2 on object 1.” • What it means: • “for every action, there is an equal and opposite reaction”

18. Newton’s Third Law • Action-Reaction Pair F21 = -F12 • Forces in nature always act in pairs • No single isolated force • The mutual actions of two bodies upon each other are ALWAYS equal and directed contrary to one another

19. Newton’s Third Law • Action-Reaction Pairs • Newton’s Law uses the forces acting on an object • n and F are both acting on the object • n is referred as to the normal force and is the force exerted by the TV stand on the TV

20. Applications of Newton’s Laws • An object in equilibrium has no net external force acting on it, and the second law, in component form, implies that   ΣFx = 0 and   ΣFy = 0 for such an object. These two equations are useful for solving problems where the object is at rest or moving at constant velocity.

21. Applications of Newton’s Laws • An object under acceleration requires the same two equations, but with the acceleration terms included: ΣFx = max and   ΣFy = may

22. Applications of Newton’s Laws • Assumptions • Objects behave as particles • can ignore rotational motion (for now) • Masses of strings or ropes are negligible • No stretching – constant length • Interested only in the forces acting on the object • can neglect reaction forces • Pulleys are massless and frictionless • Used to change direction

23. Applications of Newton’s Laws • Free Body Diagram Diagram representing all the forces applied to an object • Represent the object as a dot • Identify all the forces acting on the object, not exerted • Choose appropriate coordinate system • Incorrect FBD means incorrect solution

24. Applications of Newton’s Laws • The force T is the tension acting on the box • n and F are the forces exerted by the earth and the ground • Only forces acting directly on the object are included in the free body diagram • Reaction forces act on other objects and so are not included

25. Newton’s Third Law n • Free Body Diagram ΣFy = N – Fg = n – mg = may but the TV is not moving, so ay = 0 n – mg = 0 Therefore, n = mg Fg

26. Solving Newton’s Second Law Problems • Read the problem at least once • Draw a picture of the system • Identify the object of primary interest • Indicate forces with arrows • Label each force • Use labels that bring to mind the physical quantity involved • Draw a free body diagram • If additional objects are involved, draw separate free body diagrams for each object • Choose a convenient coordinate system for each object • Apply Newton’s Second Law • The x- and y-components should be taken from the vector equation and written separately • Solve for the unknown(s)

27. ConcepTest 4.11On an Incline A B 1) case A 2) case B 3) both the same (N = mg) 4) both the same (0 < N < mg) 5) both the same (N = 0) Consider two identical blocks, one resting on a flat surface, and the other resting on an incline. For which case is the normal force greater?

28. ConcepTest 4.11On an Incline A B 1) case A 2) case B 3) both the same (N = mg) 4) both the same (0 < N < mg) 5) both the same (N = 0) Consider two identical blocks, one resting on a flat surface, and the other resting on an incline. For which case is the normal force greater? In Case A, we know that N = W. In Case B, due to the angle of the incline, N < W. In fact, we can see that N = W cos(q). y x N f q Wy W q

29. Newton’s Law review • 1st – Law of Inertia • Objects at rest or motion will stay that way unless acted on by a force • Sailboat • Target • 2nd – F = ma • The acceleration is proportional to the force and inversely proportional to the mass • 3rd – Action-Reaction • if a force is acted on to an object, the object will exert an equal and opposite force • Recoil

30. ConcepTest 4.1cNewton’s First Law 1) a net force acted on it 2) no net force acted on it 3) it remained at rest 4) it did not move, but only seemed to 5) gravity briefly stopped acting on it You put your book on the bus seat next to you. When the bus stops suddenly, the book slides forward off the seat. Why?

31. ConcepTest 4.1cNewton’s First Law 1) a net force acted on it 2) no net force acted on it 3) it remained at rest 4) it did not move, but only seemed to 5) gravity briefly stopped acting on it The book was initially moving forward (since it was on a moving bus). When the bus stopped, the book continued moving forward, which was its initial state of motion, and therefore it slid forward off the seat. Follow-up: What is the force that usually keeps the book on the seat? You put your book on the bus seat next to you. When the bus stops suddenly, the book slides forward off the seat. Why?

32. ConcepTest 4.9bGoing Up II a m 1) N > mg 2) N = mg 3) N < mg (but not zero) 4) N = 0 5) depends on the size of the elevator A block of mass m rests on the floor of an elevator that is accelerating upward. What is the relationship between the force due to gravity and the normal force on the block?

33. ConcepTest 4.9bGoing Up II N a > 0 m mg 1) N > mg 2) N = mg 3) N < mg (but not zero) 4) N = 0 5) depends on the size of the elevator The block is accelerating upward, so it must have a net upward force. The forces on it are N (up) and mg (down), so N must be greater than mg in order to give the net upward force! S F = N–mg = ma > 0 \N > mg Follow-up: What is the normal force if the elevator is in free fall downward? A block of mass m rests on the floor of an elevator that is accelerating upward. What is the relationship between the force due to gravity and the normal force on the block?

34. Applications of Newton’s Laws • Equilibrium • An object either at rest or moving with a constant velocity is said to be in equilibrium • The net force acting on the object is zero (since the acceleration is zero) ΣF = 0 • Should use components ΣFx = 0 ΣFy = 0

35. Applications of Newton’s Laws • Equilibrium Example - FBD

36. Applications of Newton’s Laws • Equilibrium Example – FBD • Choose the coordinate system with x along the incline and y perpendicular to the incline • Replace the force of gravity with its components

37. Applications of Newton’s Laws • Equilibrium Example?

38. Applications of Newton’s Laws • Equilibrium – Multiple Objects • When you have more than one object, the problem-solving strategy is applied to each object • Draw free body diagrams for each object • Apply Newton’s Laws to each object • Solve the equations

39. Applications of Newton’s Laws • Example • A traffic light weighing 1.00x102 N hangs from a vertical cable tied to two other cables that are fastened to a support, as in the figure to the right. The upper cables makes angles of 37° and 53° with the horizontal. Find the tension in each of the three cables.

40. Applications of Newton’s Laws • Example: An object with a mass m1 = 5.00kg rests on a frictionless horizontal table and is connected to a cable that passes over a pulley and is then fastened to a hanging object with mass m2 = 10.0 kg, as shown in Figure P4.30. Find the acceleration of each object and the tension in the table.

41. Forces of Friction • An object moving on a surface encounters resistance as it interacts through its surroundings. This resistance is called friction. • Friction is a force • The force of friction is opposite of motion • Two types of friction – static and kinetic • Friction is proportional to the normal force • Friction Examples • Car on the road

42. Forces of Friction • Static Friction, ƒs • Static friction acts to keep the object from moving • If F increases, so does ƒs • If F decreases, so does ƒs • ƒs µ n

43. Forces of Friction • Kinetic Friction, ƒk • The force of kinetic friction acts when the object is in motion • ƒk = µ n • Variations of the coefficient with speed will be ignored

44. Forces of Friction • Friction Examples

45. Forces of Friction • Friction Example • Axes are rotated as usual on an incline • The direction of impending motion would be down the plane • Friction acts up the plane • Opposes the motion • Apply Newton’s Laws and solve equations

46. Static Friction... • We want to know how it acts in fixed or “static” systems: the force provided by friction depends on the forces applied on the system (magnitude: fs≤msN) • Opposes motion that would occur if ms were zero y N Fapplied x fS mg

47. Static Friction... • If a = 0. x :FappliedfS = 0 y:N = mg • While the block is static:fS Fapplied (unlike kinetic friction) y N Fapplied x fS mg

48. Static Friction... • The maximum possible force that the friction between two objects can provide is fMAX = SN, where sis the “coefficient of static friction”. • SofSS N. • As one increases F, fS gets bigger until fS=SNand the object “breaks loose” and starts to move. y N F x fS mg

49. Static Friction... y N FMAX x Smg mg • S is discovered by increasing F until the block starts to slide: x :FMAXSN = 0 y :N = mg SFMAX / mg

50. Additional comments on Friction: • The force of friction does not depend on the area of the surfaces in contact (a relatively good approximation if there is little surface deformation) • Generally S > Kfor any system