Chapter 4Chapter 4

Force; Newton’s Laws of Force; Newton’s Laws of MotionMotion

Classical MechanicsClassical Mechanics

Describes the relationship between Describes the relationship between the motion of objects in our everyday the motion of objects in our everyday world and the forces acting on themworld and the forces acting on them

Conditions when Classical Mechanics Conditions when Classical Mechanics does not applydoes not apply• very tiny objects (< atomic sizes)very tiny objects (< atomic sizes)• objects moving near the speed of lightobjects moving near the speed of light

ForcesForces

Usually think of a force as a push or pullUsually think of a force as a push or pull Vector quantityVector quantity May be a May be a contact forcecontact force or a or a field forcefield force

• Contact forces result from physical contact Contact forces result from physical contact between two objects: pushing, pullingbetween two objects: pushing, pulling

• Field forces act between disconnected objectsField forces act between disconnected objects Also called “action at a distance”Also called “action at a distance” Gravitational force: weight of objectGravitational force: weight of object

Contact and Field ForcesContact and Field Forces

Force as vectorForce as vector

Magnitude + DirectionMagnitude + Direction Components FComponents Fx, x, FFyy

Units: Newton (N), Units: Newton (N), pound(lb)pound(lb)

1lb=4.45N1lb=4.45N

F

sin

cos

FF

FF

y

x

22

tan

yx

x

y

FFF

F

F

F

yF

xF

x

y

Addition of ForcesAddition of Forces

Tail-to tip methodTail-to tip method

Parallelogram methodParallelogram method

Components methodComponents method

A

BAC

B

A

B

A

B

ExamplesExamples

Parallel forcesParallel forces

FF11=150N, F=150N, F22=100N and 53° to F=100N and 53° to F11

Newton’s First LawNewton’s First Law

An object moves with a velocity that An object moves with a velocity that is constant in magnitude and is constant in magnitude and direction, unless acted on by a direction, unless acted on by a nonzero net forcenonzero net force• The net force is defined as the vector The net force is defined as the vector

sum of all the external forces exerted on sum of all the external forces exerted on the objectthe object

External and Internal ForcesExternal and Internal Forces

External forceExternal force• Any force that results from the Any force that results from the

interaction between the object and its interaction between the object and its environmentenvironment

Internal forcesInternal forces• Forces that originate within the object Forces that originate within the object

itselfitself• They cannot change the object’s They cannot change the object’s

velocityvelocity

InertiaInertia

Is the tendency of an object to Is the tendency of an object to continue in its original motioncontinue in its original motion

MassMass

A measure of the resistance of an A measure of the resistance of an object to changes in its motion due object to changes in its motion due to a forceto a force

Scalar quantityScalar quantity SI units are kgSI units are kg

Condition for EquilibriumCondition for Equilibrium

Net force vanishesNet force vanishes No motionNo motion

0...321 FFFF

0...321 xxxx FFFF

0...321 yyyy FFFF

Newton’s Second LawNewton’s Second Law

The acceleration of an object is The acceleration of an object is directly proportional to the net force directly proportional to the net force acting on it and inversely acting on it and inversely proportional to its mass.proportional to its mass.

• F and a are both vectorsF and a are both vectors

amF

Units of ForceUnits of Force

SI unit of force is a Newton (N)SI unit of force is a Newton (N)

US Customary unit of force is a US Customary unit of force is a pound (lb)pound (lb)• 1 N = 0.225 lb1 N = 0.225 lb

2s

mkg1N1

Sir Isaac NewtonSir Isaac Newton

1642 – 17271642 – 1727 Formulated basic Formulated basic

concepts and laws concepts and laws of mechanicsof mechanics

Universal Universal GravitationGravitation

CalculusCalculus Light and opticsLight and optics

WeightWeight

Falling objectFalling object

Weight w=mgWeight w=mg Object on a table?Object on a table?

mgF

WeightWeight

The magnitude of the gravitational The magnitude of the gravitational force acting on an object of mass force acting on an object of mass mm near the Earth’s surface is called the near the Earth’s surface is called the weight weight ww of the object of the object• w = m gw = m g is a special case of Newton’s is a special case of Newton’s

Second LawSecond Law gg is the acceleration due to gravity is the acceleration due to gravity

g g can also be found from the Law of can also be found from the Law of Universal GravitationUniversal Gravitation

More about weightMore about weight

Weight is Weight is notnot an inherent property of an inherent property of an object an object • mass mass isis an inherent property an inherent property

Weight depends upon locationWeight depends upon location

Newton’s Third LawNewton’s Third Law

If object 1 and object 2 interact, the If object 1 and object 2 interact, the force exerted by object 1 on object 2 force exerted by object 1 on object 2 is equal in magnitude but opposite in is equal in magnitude but opposite in direction to the force exerted by direction to the force exerted by object 2 on object 1.object 2 on object 1.•

• Equivalent to saying a single isolated Equivalent to saying a single isolated force cannot existforce cannot exist

12 21F F

Newton’s Third Law cont.Newton’s Third Law cont.

FF1212 may be called the may be called the actionaction force and F force and F2121 the the reactionreaction force force• Actually, either force Actually, either force

can be the action or can be the action or the reaction forcethe reaction force

The action and The action and reaction forces act reaction forces act on on differentdifferent objects objects

Example: Force TableExample: Force Table

Three forces in equilibrium!Three forces in equilibrium!

Find tension in each cable supporting Find tension in each cable supporting the 600N signthe 600N sign

ExamplesExamples

Calculate the acceleration of the boxCalculate the acceleration of the box

A 50 kg box is pulled across a 16m A 50 kg box is pulled across a 16m drivewaydriveway

Projectile Motion(Ch. 5)Projectile Motion(Ch. 5)

Example of motion in 2-dimExample of motion in 2-dim An object may move in both the x An object may move in both the x

and y directions simultaneouslyand y directions simultaneously• It moves in two dimensionsIt moves in two dimensions

The form of two dimensional motion The form of two dimensional motion we will deal with is called we will deal with is called projectile projectile motionmotion

vvo o and and o o

Assumptions of Projectile Assumptions of Projectile MotionMotion

We may ignore air frictionWe may ignore air friction We may ignore the rotation of the We may ignore the rotation of the

earthearth With these assumptions, an object in With these assumptions, an object in

projectile motion will follow a projectile motion will follow a parabolic pathparabolic path

Rules of Projectile MotionRules of Projectile Motion

The horizontal motion (x) and vertical of The horizontal motion (x) and vertical of motion (y) are completely independent of motion (y) are completely independent of each othereach other

The x-direction is uniform motionThe x-direction is uniform motion• aaxx = 0 = 0

The y-direction is free fallThe y-direction is free fall• aayy = -g = -g

The initial velocity can be broken down The initial velocity can be broken down into its x- and y-componentsinto its x- and y-components• cos sinOx O O Oy O Ov v v v

Projectile MotionProjectile Motion

Projectile Motion at Various Projectile Motion at Various Initial AnglesInitial Angles

Complementary Complementary values of the values of the initial angle result initial angle result in the same rangein the same range• The heights will be The heights will be

differentdifferent The maximum The maximum

range occurs at a range occurs at a projection angle projection angle of 45of 45oo

Some Details About the RulesSome Details About the Rules

Horizontal motion Horizontal motion • vvx x =v=vxo xo =v=vo o coscosoo

• x = xx = xoo+v+vxoxott

This is the only operative equation in the x-This is the only operative equation in the x-direction since there is uniform velocity in direction since there is uniform velocity in that directionthat direction

constantvcosvv xooxo

More Details About the RulesMore Details About the Rules

Vertical motion-- free fall problemVertical motion-- free fall problem• vvyy=v=voosinsinoo-gt-gt

• y=y=yoo+ (v+ (voosinsinoo)t-(1/2)gt)t-(1/2)gt22

• the positive direction as upwardthe positive direction as upward• uniformly accelerated motion, so the uniformly accelerated motion, so the

motion equations all holdmotion equations all hold

ooyo sinvv

Velocity of the ProjectileVelocity of the Projectile

The velocity of the projectile at any The velocity of the projectile at any point of its motion is the vector sum point of its motion is the vector sum of its x and y components at that of its x and y components at that pointpoint

• Remember to be careful about the Remember to be careful about the angle’s quadrantangle’s quadrant

2 2 1tan yx y

x

vv v v and

v

ExampleExample

Superman hits a home run with Superman hits a home run with vvoo=15m/s and =15m/s and oo=60°=60°

Some useful resultsSome useful results

TrajectoryTrajectory

Maximum HeightMaximum Height

RangeRange

222 cos2

)(tan xv

gxy

ooo

g

vh oo

2

sin 22

g

vR oo

)2sin(2

ExampleExample

A daredevil jumps a canyon 15m wide A daredevil jumps a canyon 15m wide by driving a motorcycle up an incline by driving a motorcycle up an incline sloped at an angle of 37° with the sloped at an angle of 37° with the horizontal. What minimum speed horizontal. What minimum speed must she have in order to clear the must she have in order to clear the canyon? How long will she be in the canyon? How long will she be in the air?air?