Chapters 4, 5 Force and Laws of Motion

1. Chapters 4, 5 Force and Laws of Motion

2. What causes motion? • That’s the wrong question! • The ancient Greek philosopher Aristotle believed that forces - pushes and pulls - caused motion • The Aristotelian view prevailed for some 2000 years • Galileo first discovered the correct relation between force and motion • Force causes not motion itself but change in motion Aristotle (384 BC – 322 BC) Galileo Galilei (1564 – 1642)

3. Sir Isaac Newton (1643 – 1727) • Newtonian mechanics • Describes motion and interaction of objects • Applicable for speeds much slower than the speed of light • Applicable on scales much greater than the atomic scale • Applicable for inertial reference frames – frames that don’t accelerate themselves

4. Force • What is a force? • Colloquial understanding of a force – a push or a pull • Forces can have different nature • Forces are vectors • Several forces can act on a single object at a time – they will add as vectors

5. Force superposition • Forces applied to the same object are adding as vectors – superposition • The net force – a vector sum of all the forces applied to the same object

6. Newton’s First Law • If the net force on the body is zero, the body’s acceleration is zero

7. Newton’s Second Law • If the net force on the body is not zero, the body’s acceleration is not zero • Acceleration of the body is directly proportional to the net force on the body • The coefficient of proportionality is equal to the mass (the amount of substance) of the object

8. Newton’s Second Law • SI unit of force kg*m/s2 = N (Newton) • Newton’s Second Law can be applied to all the components separately • To solve problems with Newton’s Second Law we need to consider a free-body diagram • If the system consists of more than one body, only external forces acting on the system have to be considered • Forces acting between the bodies of the system are internal and are not considered

9. Newton’s Third Law • When two bodies interact with each other, they exert forces on each other • The forces that interacting bodies exert on each other, are equal in magnitude and opposite in direction

10. Forces of different origins • Gravitational force • Normal force • Tension force • Frictional force (friction) • Drag force • Spring force

11. Gravity force (a bit of Ch. 8) • Any two (or more) massive bodies attract each other • Gravitational force (Newton's law of gravitation) • Gravitational constant G = 6.67*10 –11 N*m2/kg2 = 6.67*10 –11 m3/(kg*s2) – universal constant

12. Gravity force at the surface of the Earth g = 9.8 m/s2

13. Gravity force at the surface of the Earth • The apple is attracted by the Earth • According to the Newton’s Third Law, the Earth should be attracted by the apple with the force of the same magnitude

14. Weight • Weight (W) of a body is a force that the body exerts on a support as a result of gravity pull from the Earth • Weight at the surface of the Earth: W = mg • While the mass of a body is a constant, the weight may change under different circumstances

15. Tension force • A weightless cord (string, rope, etc.) attached to the object can pull the object • The force of the pull is tension ( T ) • The tension is pointing away from the body

16. Free-body diagrams

17. Chapter 4 Problem 56 Your engineering firm is asked to specify the maximum load for the elevators in a new building. Each elevator has mass 490 kg when empty and maximum acceleration 2.24 m/s2. The elevator cables can withstand a maximum tension of 19.5 kN before breaking. For safety, you need to ensure that the tension never exceeds two-thirds of that value. What do you specify for the maximum load? How many 70-kg people is that?

18. Normal force • When the body presses against the surface (support), the surface deforms and pushes on the body with a normal force (n) that is perpendicular to the surface • The nature of the normal force – reaction of the molecules and atoms to the deformation of material

19. Normal force • The normal force is not always equal to the gravitational force of the object

20. Free-body diagrams

21. Free-body diagrams

22. Chapter 5 Problem 19 If the left-hand slope in the figure makes a 60° angle with the horizontal, and the right-hand slope makes a 20° angle, how should the masses compare if the objects are not to slide along the frictionless slopes?

23. Spring force • Spring in the relaxed state • Spring force (restoring force) acts to restore the relaxed state from a deformed state

24. Robert Hooke(1635 – 1703) • Hooke’s law • For relatively small deformations • Spring force is proportional to the deformation and opposite in direction • k – spring constant • Spring force is a variable force • Hooke’s law can be applied not to springs only, but to all elastic materials and objects

25. Frictional force • Friction ( f) - resistance to the sliding attempt • Direction of friction – opposite to the direction of attempted sliding (along the surface) • The origin of friction – bonding between the sliding surfaces (microscopic cold-welding)

26. Static friction and kinetic friction • Moving an object: static friction vs. kinetic

27. Friction coefficient • Experiments show that friction is related to the magnitude of the normal force • Coefficient of static frictionμs • Coefficient of kinetic frictionμk • Values of the friction coefficients depend on the combination of surfaces in contact and their conditions (experimentally determined)

28. Free-body diagrams

29. Free-body diagrams

30. Chapter 5 Problem 30 Starting from rest, a skier slides 100 m down a 28° slope. How much longer does the run take if the coefficient of kinetic friction is 0.17 instead of 0?

31. Drag force • Fluid – a substance that can flow (gases, liquids) • If there is a relative motion between a fluid and a body in this fluid, the body experiences a resistance (drag) • Drag force (R) • R = ½DρAv2 • D - drag coefficient; ρ – fluid density; A– effective cross-sectional area of the body (area of a cross-section taken perpendicular to the velocity); v- speed

32. Terminal velocity • When objects falls in air, the drag force points upward (resistance to motion) • According to the Newton’s Second Law • ma = mg – R = mg – ½DρAv2 • As v grows, a decreases. At some point acceleration becomes zero, and the speed value riches maximum value – terminal speed • ½DρAvt2 = mg

33. Terminal velocity • Solving ½DρAvt2 = mg we obtain vt = 300 km/h vt = 10 km/h

34. Centripetal force • For an object in a uniform circular motion, the centripetal acceleration is • According to the Newton’s Second Law, a force must cause this acceleration – centripetal force • A centripetal force accelerates a body by changing the direction of the body’s velocity without changing the speed

35. Centripetal force • Centripetal forces may have different origins • Gravitation can be a centripetal force • Tension can be a centripetal force • Etc.

36. Centripetal force • Centripetal forces may have different origins • Gravitation can be a centripetal force • Tension can be a centripetal force • Etc.

37. Free-body diagram

38. Chapter 5 Problem 25 You’re investigating a subway accident in which a train derailed while rounding an unbanked curve of radius 132 m, and you’re asked to estimate whether the train exceeded the 45-km/h speed limit for this curve. You interview a passenger who had been standing and holding onto a strap; she noticed that an unused strap was hanging at about a 15° angle to the vertical just before the accident. What do you conclude?

39. Answers to the even-numbered problems Chapter 4 Problem 20 7.7 cm

40. Answers to the even-numbered problems Chapter 4 Problem 26 590 N

41. Answers to the even-numbered problems Chapter 4 Problem 38 5.77 N; 72.3°

42. Answers to the even-numbered problems Chapter 5 Problem 28 580 N; opposite to the motion of the cabinet

43. Answers to the even-numbered problems Chapter 5 Problem 50 110 m