Chapters 5 6 Force and Motion Newtonian mechanics

Chapters 5, 6 Force and Motion

Newtonian mechanics Sir Isaac Newton (1643 – 1727) • 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

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

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

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

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

Newton’s Second Law • SI unit of force kg*m/s 2 = 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

Chapter 5 Problem 6

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

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

Gravity force (a bit of Ch. 13) • Any two (or more) massive bodies attract each other • Gravitational force (Newton's law of gravitation) • Gravitational constant G = 6. 67*10 – 11 N*m 2/kg 2 = 6. 67*10 – 11 m 3/(kg*s 2) – universal constant

Gravity force at the surface of the Earth g = 9. 8 m/s 2

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

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

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

Free-body diagrams

Chapter 5 Problem 47

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

Free-body diagrams

Free-body diagrams

Chapter 5 Problem 41

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)

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

Friction coefficient • 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)

Free-body diagrams

Free-body diagrams

Chapter 6 Problem 23

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 (D) D = ½CρAv 2 • C - drag coefficient; ρ – fluid density; A – effective cross-sectional area of the body (area of a crosssection taken perpendicular to the velocity); v - speed

Terminal velocity • When objects falls in air, the drag force points upward (resistance to motion) • According to the Newton’s Second Law ma = mg – D = mg – ½CρAv 2 • As v grows, a decreases. At some point acceleration becomes zero, and the speed value riches maximum value – terminal speed ½CρAvt 2 = mg

Terminal velocity Solving ½CρAvt 2 = mg we obtain vt = 300 km/h vt = 10 km/h

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

Hooke’s law • For relatively small deformations Robert Hooke (1635 – 1703) • 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

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

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

Free-body diagram

Answers to the even-numbered problems Chapter 5: Problem 2. (a)1. 88 N; (b) 0. 684 N; (c) (1. 88 N)ˆi + (0. 684 N)ˆj

Answers to the even-numbered problems Chapter 5: Problem 10. (a)2. 0 N; (b) down

Answers to the even-numbered problems Chapter 5: Problem 22. (a) 5. 5 k. N; (b) 2. 7 s; (c) 4. 0; (d) 2. 0

Answers to the even-numbered problems Chapter 6: Problem 2. 0. 61

Answers to the even-numbered problems Chapter 6: Problem 32. 3. 75

Answers to the even-numbered problems Chapter 6: Problem 36. 48 km/h

Answers to the even-numbered problems Chapter 6: Problem 40. (a) 3. 7 k. N; (b) up; (c) 1. 3 k. N; (d) down

Answers to the even-numbered problems Chapter 6: Problem 104. (a)0. 13 N; (b) 0. 12