1. Force and Laws of Motion
Chapters 4, 5
Force and Laws of Motion

2. What causes motion? That’s the wrong question!
Aristotle
(384 BC – 322 BC)
Galileo Galilei
(1564 – 1642)
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

3. Newtonian mechanics Describes motion and interaction of objects
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. Hooke’s law For relatively small deformations
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 Coefficient of kinetic friction μk
Experiments show that friction is related to the magnitude of the normal force
Coefficient of static friction μs
Coefficient of kinetic friction μk
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 ma = mg – R = mg – ½DρAv2 ½DρAvt2 = mg
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

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