Chapter 4

Raymond A. Serway

Chris Vuille

Chapter Four

The Laws of Motion

Classical Mechanics

•Describes the relationship between themotion of objects in our everyday world andthe forces acting on them

•Conditions when Classical Mechanics does notapply

–Very tiny objects (< atomic sizes)

–Objects moving near the speed of light

Introduction

Newton’s Laws

•This chapter will look at an introduction toNewton’s three laws of motion and his law ofgravity

•These laws are considered among the greatestachievements of the human mind

Introduction

Forces

•Commonly imagined as a push or pull on someobject

•Vector quantity

•May be a contact force or a field force

–Contact forces result from physical contact between twoobjects

–Field forces act between disconnected objects

Section 4.1

Contact and Field Forces

Section 4.1

Fundamental Forces

•Types

–Strong nuclear force

–Electromagnetic force

–Weak nuclear force

–Gravity

•Characteristics

–All field forces

–Listed in order of decreasing strength

–Only gravity and electromagnetic in mechanics

Section 4.1

Sir Isaac Newton

•1642 – 1727

•Formulated basicconcepts and laws ofmechanics

•Universal Gravitation

•Calculus

•Light and optics

•Built upon Galileo’sconcepts of motion

Section 4.2

Newton’s First Law

•An object moves with a velocity that isconstant in magnitude and direction, unlessacted on by a nonzero net force

–The net force is defined as the vector sum of allthe external forces exerted on the object

Section 4.2

External and Internal Forces

•External force

–Any force that results from the interactionbetween the object and its environment

•Internal forces

–Forces that originate within the object itself

–They cannot change the object’s velocity

Section 4.2

Inertia

•Is the tendency of an object to continue in itsoriginal motion

–In the absence of a force

•Thought experiment

–Hit a golf ball

–Hit a bowling ball with the same force

–The golf ball will travel farther

–Both resist changes in their motion

Section 4.2

Mass

•A measure of the resistance of an object tochanges in its motion due to a force

–The larger the mass, the less it accelerates underthe action of a given force

•SI units are kg

•Scalar quantity

Section 4.2

Newton’s Second Law

•The acceleration of an object is directlyproportional to the net force acting on it andinversely proportional to its mass.

•Can also be applied three-dimensionally

Section 4.3

Units of Force

•SI unit of force is a Newton (N)

•US Customary unit of force is a pound (lb)

–1 N = 0.225 lb

•See table 4.1 for a summary of units

Section 4.3

Some Notes About Forces

•Forces cause changes in motion

–Motion can occur in the absence of forces (constantvelocity, no acceleration)

•All the forces acting on an object are added asvectors to find the net force acting on the object

–Mass is not a force itself

•Newton’s Second Law is a vector equation

Section 4.3

Gravitational Force

•Mutual force of attraction between any twoobjects

•Expressed by Newton’s Law of UniversalGravitation:

–Every particle in the Universe attracts every other particlewith a force that is directly proportional to the product ofthe masses of the particles and inversely proportional tothe square of the distance between them

Section 4.3

Weight

•The magnitude of the gravitational forceacting on an object of mass m near the Earth’ssurface is called the weight w of the object

–w = m g is a special case of Newton’s Second Law

•g is the acceleration due to gravity

•g can also be found from the Law of UniversalGravitation

Section 4.3

More about weight

•Weight is not an inherent property of anobject

–Mass is an inherent property

•Weight depends upon location (accelerationdue to gravity times mass equals weight)

Section 4.3

Newton’s Third Law

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

–

–Equivalent to saying a single isolated force cannotexist

Section 4.4

Newton’s Third Law cont.

•F12 may be called the actionforce and F21 the reactionforce

–Actually, either force can bethe action or the reactionforce

•The action and reactionforces act on differentobjects

Section 4.4

Some Action-Reaction Pairs

•

– is the normal force, theforce the table exerts on theTV

– is always perpendicular tothe surface

– is the reaction – the TV onthe table

–

Section 4.4

More Action-Reaction pairs

•

– is the force the Earthexerts on the object

– is the force the objectexerts on the earth

–

Section 4.4

•Newton’s Law uses theforces acting on anobject

• are acting onthe object

• are acting onother objects

Forces Acting on an Object

Section 4.4

Applications of Newton’s Laws

•Assumptions

–Objects behave as particles

•Can ignore rotational motion (for now)

–Masses of strings or ropes are negligible

–Interested only in the forces acting on the object

•Can neglect reaction forces

Section 4.5

More Assumptions – Ropes

•Ignore any frictional effects of the rope

•Ignore the mass of the rope

•The magnitude of the force exerted along the rope is called thetension

•The tension is the same at all points in the rope

Section 4.5

Free Body Diagram

•A diagram of the forces acting on an object

•Must identify all the forces acting on theobject of interest

•Choose an appropriate coordinate system

•If the free body diagram is incorrect, thesolution will likely be incorrect

Section 4.5

Free Body Diagram, Example

•The magnitude of force T isthe tension acting on thebox

–The tension is the same at allpoints along the rope

• are the forcesexerted by the earth andthe ground

Section 4.5

Free Body Diagram, final

•Only forces acting directly on the object areincluded in the free body diagram

–Reaction forces act on other objects and so arenot included

–The reaction forces do not directly influence theobject’s motion

Section 4.5

Solving Newton’s Second LawProblems

•Read the problem at least once

•Draw a picture of the system

–Identify the object of primary interest

–Indicate forces with arrows

•Label each force

–Use labels that bring to mind the physical quantityinvolved

Section 4.5

Solving Newton’s Second LawProblems, cont

•Draw a free body diagram

–If additional objects are involved, draw separate free bodydiagrams for each object

–Choose a convenient coordinate system for each object

•Apply Newton’s Second Law

–The x- and y-components should be taken from the vectorequation and written separately

•Solve for the unknown(s)

Section 4.5

Equilibrium

•An object either at rest or moving with aconstant velocity is said to be in equilibrium

•The net force acting on the object is zero(since the acceleration is zero)

Section 4.5

Equilibrium cont.

•Easier to work with the equation in terms of itscomponents:

•This could be extended to three dimensions

•A zero net force does not mean the object is notmoving, but that it is not accelerating

Section 4.5

Equilibrium Example – Free BodyDiagrams

Section 4.5

Inclined Planes

•Choose the coordinatesystem with x along theincline and yperpendicular to theincline

•Replace the force ofgravity with itscomponents

Section 4.5

Multiple Objects – Example

•When you have more than one object, theproblem-solving strategy is applied to eachobject

•Draw free body diagrams for each object

•Apply Newton’s Laws to each object

•Solve the equations

Section 4.5

Multiple Objects – Example, cont.

Section 4.5

Note: The magnitude of the tension

force is the same throughout the

rope, and opposes gravity for each

object.

Forces of Friction

•When an object is in motion on a surface orthrough a viscous medium, there will be aresistance to the motion

–This is due to the interactions between the objectand its environment

•This is resistance is called friction

Section 4.6

More About Friction

•Friction is proportional to the normal force

•The force of static friction is generally greater thanthe force of kinetic friction

•The coefficient of friction (µ) depends on thesurfaces in contact

•The direction of the frictional force is opposite thedirection of motion

•The coefficients of friction are nearly independent ofthe area of contact

Section 4.5

Static Friction, ƒs

•Static friction acts tokeep the object frommoving

•If F increases, so does ƒs

•If F decreases, so does ƒs

•ƒs  µs n

–Use = sign for impendingmotion only

Section 4.5

Kinetic Friction, ƒk

•The force of kineticfriction acts when theobject is in motion

•ƒk = µk n

–Variations of thecoefficient with speedwill be ignored

Section 4.5

Friction, final

•Adjust the force and seewhere you are on thegraph

•Note especially where ƒ= Fs

Section 4.5

Some Coefficients of Friction

Section 4.5

Block on a Ramp, Example withFriction

•Axes are rotated as usual onan incline

•The direction of impendingmotion would be down theplane

•Friction acts up the plane

–Opposes the motion

•Apply Newton’s Laws andsolve equations

Section 4.5

Example

•How fast is a block traveling on a frictionlessramp 25 degrees above the horizontal if theblock is released from rest, and allowed totravel for 10.0 meters down the ramp?

Connected Objects

•Apply Newton’s Lawsseparately to each object

•The magnitude of theacceleration of both objectswill be the same

•The tension is the same ineach diagram

•Solve the simultaneousequations

Section 4.5

Connected Objects – SystemApproach

•Treating the system as one object allows analternative method or a check

–Use only external forces

•Not the tension – it’s internal

–The mass is the mass of the system

•Doesn’t tell you anything about any internalforces

Section 4.5

Example

Masses m1 = 4.00 kg and m2 = 9.00 kg are connected by a light string that passes overa frictionless pulley. As shown in the diagram, m1 is held at rest on the floor and m2 restson a fixed incline of angle 40 degrees. The masses are released from rest, and m2 slides1.00 m down the incline in 4 seconds. Determine (a) The acceleration of each mass (b)The coefficient of kinetic friction and (c) the tension in the string.

m1g

m2g