#### Transcript Chapters 5-6

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 Fnet 0 a 0 Newton’s Second Law • If the net force on the body is not zero, the body’s acceleration is not zero Fnet 0 a 0 • 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 ma Fnet Fnet a m 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 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 FBC FCB 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) m1m2 F G 2 rˆ r • Gravitational constant G = 6.67*10 –11 N*m2/kg2 = 6.67*10 –11 m3/(kg*s2) – universal constant Gravity force at the surface of the Earth mEarthmCrate ˆ m1m2 FCrate G 2 rˆ G j 2 r REarth GmEarth mCrate ˆj g mCrate ˆj FCrate 2 REarth g = 9.8 m/s2 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 mEarthmApple m1m2 ˆj FEarth G 2 rˆ G 2 r REarth a Earth G mEarthm Apple 2 Earth R mEarth m Apple GmEarth mApple ˆ ˆj ˆj g j R2 m mEarth Earth Earth 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 f s ,max s FN • Coefficient of kinetic friction μk f k k FN • 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 = ½CρAv2 • 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 – ½CρAv2 • As v grows, a decreases. At some point acceleration becomes zero, and the speed value riches maximum value – terminal speed ½CρAvt2 = mg Terminal velocity Solving ½CρAvt2 = mg we obtain 2mg vt C A 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 Fs kd 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 2 v ac R • According to the Newton’s Second Law, a force must cause this acceleration – centripetal force mv Fc mac R 2 • 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)2.0N; (b) down Answers to the even-numbered problems Chapter 5: Problem 22. (a) 5.5 kN; (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. 48km/h Answers to the even-numbered problems Chapter 6: Problem 40. (a) 3.7 kN; (b) up; (c) 1.3 kN; (d) down Answers to the even-numbered problems Chapter 6: Problem 104. (a)0.13 N; (b) 0.12