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

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

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

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/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

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. 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

Gravity force at the surface of the Earth 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

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 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?

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

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

Free-body diagrams

Free-body diagrams

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?

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) 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

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 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)

Free-body diagrams

Free-body diagrams

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?

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

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

Terminal velocity Solving ½DρAvt2 = mg we obtain vt = 300 km/h

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.

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

Free-body diagram

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?

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

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

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

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

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