CHAPTER 12CHAPTER 12

Kinetics of Particles:Kinetics of Particles:

Newton’s Second LawNewton’s Second Law

Reading AssignmentReading Assignment

12.1 INTRODUCTION12.1 INTRODUCTION

12.2 NEWTON’S SECOND LAW OF 12.2 NEWTON’S SECOND LAW OF MOTIONMOTION

If the resultant force acting on a particle If the resultant force acting on a particle is not zero, the particle will have an is not zero, the particle will have an acceleration proportional to the acceleration proportional to the magnitude of the resultant and in the magnitude of the resultant and in the direction of this resultant force.direction of this resultant force.

amF

amF

CauseCause= Effect= Effect

More accuratelyMore accurately

Inertial frame Inertial frame

or Newtonian frame of reference – or Newtonian frame of reference –

one in which Newton’s second law one in which Newton’s second law equation holds. equation holds. Wikipedia definition..

Free Body Diagrams (FBD)Free Body Diagrams (FBD)

This is a diagram showing some object and This is a diagram showing some object and the forces applied to it.the forces applied to it.

It contains only forces and coordinate It contains only forces and coordinate information, nothing else.information, nothing else.

There are only two kinds of forces to be There are only two kinds of forces to be considered in mechanics:considered in mechanics:

Force of gravityForce of gravity

Contact forcesContact forces

x

y

Example FBDExample FBDA car of mass m rests on a 30A car of mass m rests on a 3000 incline. incline.

mmgg

NN

FF

FBDFBD

This completes the FBD.This completes the FBD.

Example FBDExample FBDA car of mass m rests on a 30A car of mass m rests on a 3000 incline. incline.

mmgg

NN

FF

FBDFBD

Just for grins, let’s doJust for grins, let’s doa vector addition.a vector addition. x

y

yma

sinmg

Newton’s Second LawNewton’s Second LawNSLNSL

A car of mass m rests on a 30A car of mass m rests on a 3000 incline. incline.

mg

N

F

NSL

xxmaF

yymaF

x

y

F xma

N cosmg

What if friction is smaller?What if friction is smaller?

FBDFBD

Newton’s Second LawNewton’s Second LawNSLNSL

A car of mass m rests on a 30A car of mass m rests on a 3000 incline. incline.

mg

N

F

NSL

xxmaF

xmasinmg F

yymaF

ymacosmgN

oops

x

y

12.3 LINEAR MOMENTUM OF A PARTICLE. 12.3 LINEAR MOMENTUM OF A PARTICLE. RATE OF CHANGE OF LINEAR RATE OF CHANGE OF LINEAR

MOMENTUMMOMENTUM

dt

vdmF

)vm(

dt

d

LdtLdF

MomentumLinearvmL

Linear Momentum Conservation Linear Momentum Conservation PrinciplePrinciple: : If the resultant force on a particle is If the resultant force on a particle is zero, the linear momentum of the zero, the linear momentum of the particle remains constant in both particle remains constant in both magnitude and direction.magnitude and direction.

0LF

ttanConsL

12.4 SYSTEMS OF UNITS12.4 SYSTEMS OF UNITS

Reading AssignmentReading Assignment

12.5 EQUATIONS OF MOTION12.5 EQUATIONS OF MOTION

Rectangular ComponentsRectangular Components

)kajaia(m)kFjFiF( zyxzyx

oror

xx maF

yy maF

zz maF zm ym

xm

x

z

y

O

For Projectile MotionFor Projectile Motion

0xm Wym 0zm

0x gm

Wy

0z

In the x-y planeIn the x-y plane

tF

nF

nam

tam

Tangential and Normal Components Tangential and Normal Components

x

z

y

O

mnn maF tt maF

dtdvmFt

2

nvmF

And as a reminderAnd as a reminder

Take Newton’s second Take Newton’s second law,law,

This has the appearance of This has the appearance of being in static equilibrium and is being in static equilibrium and is actually referred to as actually referred to as dynamic dynamic equilibriumequilibrium..

12.6 DYNAMIC EQUILIBRIUM12.6 DYNAMIC EQUILIBRIUM

Don’t ever use this method in my course … H. DowningDon’t ever use this method in my course … H. Downing

amF

0 amF

vm

x

z

y

O

r

OH

12.7 ANGULAR MOMENTUM OF A 12.7 ANGULAR MOMENTUM OF A PARTICLE. RATE OF CHANGE OF PARTICLE. RATE OF CHANGE OF

ANGULAR MOMENTUMANGULAR MOMENTUM

vmrH0

sinrmvH0

Angular Momentum of a ParticleAngular Momentum of a Particle

moment of momentum or the moment of momentum or the angular momentum of the angular momentum of the particle about particle about OO. It is . It is perpendicular to the plane perpendicular to the plane containing the position vector containing the position vector and the velocity vector.and the velocity vector.

vmrH0

zyx

0

mvmvmv

zyx

kji

H

)zvyv(mH yzx

)yvxv(mHxyz

)xvzv(mH zxy

In Polar In Polar Coordinates Coordinates

x

y

vm

O

r

vm

rvm

For motion in For motion in x-yx-y plane plane

)yvxv(mHxyz

sinrmvH0 rmv 2mr

0)zvyv(mHyzx

0)xvzv(mHzxy

amrvmv

00 MH

vmrvmrH0 amr

Derivative of angular momentum with respect to Derivative of angular momentum with respect to time,time,

12.8 EQUATIONS OF MOTION IN TERMS 12.8 EQUATIONS OF MOTION IN TERMS OF RADIAL AND TRANSVERSE OF RADIAL AND TRANSVERSE

COMPONENTSCOMPONENTS

Consider particle at Consider particle at rr and and , in polar , in polar coordinates,coordinates,

This latter result may also beThis latter result may also be

derived from angular momentum.derived from angular momentum.

x

y

O

r

F

rF

2rr rrmmaF

r2rmmaF

This latter result may also beThis latter result may also be

derived from angular momentum.derived from angular momentum.

x

y

O

r

F

rF

2rr rrmmaF

r2rmmaF

2O mrH

r2rmF

2mr

dt

dFr

rr2rm 2

12.9 MOTION UNDER A CENTRAL 12.9 MOTION UNDER A CENTRAL FORCE. CONSERVATION OF FORCE. CONSERVATION OF

ANGULAR MOMENTUMANGULAR MOMENTUM

xO

y

mF

When the only force acting on particle is directed When the only force acting on particle is directed toward or away from a fixed point toward or away from a fixed point O, the particle is , the particle is said to be said to be moving under a central forcemoving under a central force..

Since the line of action of the central Since the line of action of the central force passes through force passes through O,,

0HMOO

Position vector and motion of particle are Position vector and motion of particle are in a plane perpendicular toin a plane perpendicular to

OH

x

y

O

mF

0H0 ttanConsH

vmr

Position vector and motion of Position vector and motion of particle are in a plane particle are in a plane perpendicular toperpendicular to

OH

Since the angular momentum is Since the angular momentum is constant, its magnitude can be written constant, its magnitude can be written asas

sinrmvsinmvr000

RememberRemember

o

2 Hmr hrmH 20

Conservation of Angular MomentumConservation of Angular Momentum• Radius vector Radius vector OPOP sweeps sweeps

infinitesimal areainfinitesimal area

drdA 221

• Areal velocityAreal velocity

2

212

21 r

dtdr

dtdA

• Recall, for a body moving under a Recall, for a body moving under a central force,central force,

constant2 rh• When a particle moves under a central When a particle moves under a central

force, its areal velocity is constant.force, its areal velocity is constant.

x

y

O

F

r

d

dArd

P

22 s/ft2.32s/m81.9g mgr

GMmW

2

228 sl/ftlb1044.3 4

49

slb

ft104.34

2211 kg/mN1067.6

2

312

skg

m107.66

egrationintofttanconsG 2r

MmGF

12.10 NEWTON’S LAW OF GRAVITATION12.10 NEWTON’S LAW OF GRAVITATION

• Gravitational force exerted by the sun on a Gravitational force exerted by the sun on a planet or by the earth on a satellite is an planet or by the earth on a satellite is an important example of gravitational force.important example of gravitational force.

• Newton’s law of universal gravitationNewton’s law of universal gravitation - two - two particles of mass particles of mass MM and and mm attract each other attract each other with equal and opposite forces directed with equal and opposite forces directed along the line connecting the particles,along the line connecting the particles,

• For particle of mass For particle of mass mm on the earth’s on the earth’s surface,surface,

F

F

-

m

M

r

12.11 TRAJECTORY OF A PARTICLE 12.11 TRAJECTORY OF A PARTICLE UNDER A CENTRAL FORCEUNDER A CENTRAL FORCE

For particle moving under central force directed towards force For particle moving under central force directed towards force center,center,

Second expression is Second expression is equivalent toequivalent to

, constanthr 2

2r

h

r2 Frrm F

Fr2rm 0

dt

drr

dt

d

d

dr

d

dr

r

h2

2r

h

r

1

d

dh

Remember that Remember that

dt

rdr

d

rd

r

h2

r

1

d

d

r

h2

2

2

2

r1

u LetLet

2

222

d

uduhr

Frrm 2 Remember thatRemember that

Frrm 2

2

222

d

uduhr

Fuhd

uduhm 32

2

222

222

2

umh

Fu

d

ud

This can be solved, sometimes.This can be solved, sometimes.

If If F is a known function of is a known function of r or or u,, then particle trajectory then particle trajectory may be found by integrating for may be found by integrating for u = f(), with constants , with constants of integration determined from initial conditions.of integration determined from initial conditions.

12.12 APPLICATION TO SPACE 12.12 APPLICATION TO SPACE MECHANICSMECHANICS

Consider earth satellites subjected to Consider earth satellites subjected to only gravitational pull of the earth.only gravitational pull of the earth.

2r

GMmF 2GMmu

222

2

umh

Fu

d

ud

22

2

h

GMu

d

ud

OA

r

22

2

h

GMu

d

ud

There are two solutions:There are two solutions:General SolutionGeneral SolutionParticular SolutionParticular Solution

)cos(Cu 0 2h

GM r

1

From the figure choose polar axis so thatFrom the figure choose polar axis so that

00

The above equation for The above equation for u is a conic section, is a conic section,that is it is the equation forthat is it is the equation for

ellipses (and circles), parabolas, and hyperbolas.ellipses (and circles), parabolas, and hyperbolas.

OA

r

CircleCircle EllipseEllipse ParabolaParabola HyperbolaHyperbola

Conic SectionsConic Sections

EccentricityEccentricity2h

GMC

cos1h

GM2

Origin, located at earth’s center, Origin, located at earth’s center,

is a focus of the conic section.is a focus of the conic section.

Trajectory may be ellipse, parabola, or hyperbola Trajectory may be ellipse, parabola, or hyperbola depending on value of eccentricity.depending on value of eccentricity.

)cos(Cu 0 2h

GM r

1

OA

r

cosCh

GMr1

2

OA

r

1 cos1

hGM

r1

2

Hyperbola, Hyperbola, > 1 or or C > GM/h2..

The radius vector becomes The radius vector becomes infinite forinfinite for

0cos11

1cos 11

2

1

hCGMcos

O A

r

1 cos1

hGM

r1

2

0cos12

0

2180

Parabola, Parabola, = 1 or or C = GM/h2..

The radius vector becomes The radius vector becomes infinite forinfinite for

1

cos1h

GMr1

2

Ellipse, Ellipse, < 1 or or C < GM/h2. .

The radius vector is finite for The radius vector is finite for all all , and is constant for a , and is constant for a circle, for circle, for = 0..

OA

r

OA

LaunchingLaunching

Powered FlightPowered Flight

0v

BurnoutBurnout0r

hvr00

andand

2

0

2

0

2 vrGM

hGM

2

0

2

0

2

vrgR

Ch

GMr1

20

20 h

GMr1

C

Integration constant Integration constant C is is determined by conditions determined by conditions at beginning of free flight, at beginning of free flight, =0, , r = r0 ..

OA

LaunchingLaunchingPowered FlightPowered Flight

0v

BurnoutBurnout0r

Ch

GMr1

2

0

Escape VelocityEscape Velocity

2hGMC1

Remember thatRemember that

ForFor

2hGM

C

2

0

2

0

2

0vr

GM2hGM2

r1

0

0 rGM2v

OA

r

1

If the initial velocity is less than If the initial velocity is less than the escape velocity, the satellite the escape velocity, the satellite will move in elliptical orbits.will move in elliptical orbits.If If = 0, then, then

2

0h

GMr1

0

hGM

C2

2

0

2

0

2

0vr

GMh

GMr1

0circle r

GMv

C

O AA’ O’ C

B

r1 r0

a

b

Recall that for a particle moving under a Recall that for a particle moving under a central force, the central force, the areal velocityareal velocity is is constant, i.e.,constant, i.e.,

constanthrdtdA

212

21

Periodic timePeriodic time or time or time required for a satellite to required for a satellite to complete an orbit is equal complete an orbit is equal to the area within the orbit to the area within the orbit divided by divided by areal velocityareal velocity,,

h

ab2

2h

ab

102

1 rra WhereWhere

10rrb

12.13 KEPLER’S LAWS OF PLANETARY 12.13 KEPLER’S LAWS OF PLANETARY MOTIONMOTION

• Results obtained for trajectories of satellites around earth Results obtained for trajectories of satellites around earth may also be applied to trajectories of planets around the may also be applied to trajectories of planets around the sun.sun.

• Properties of planetary orbits around the sun were Properties of planetary orbits around the sun were determined by astronomical observations by Johann determined by astronomical observations by Johann Kepler (1571-1630) before Newton had developed his Kepler (1571-1630) before Newton had developed his fundamental theory.fundamental theory.

1)1) Each planet describes an ellipse, with the sun Each planet describes an ellipse, with the sun located at one of its foci.located at one of its foci.

2)2) The radius vector drawn from the sun to a planet The radius vector drawn from the sun to a planet sweeps equal areas in equal times.sweeps equal areas in equal times.

3)3) The squares of the periodic times of the planets are The squares of the periodic times of the planets are proportional to the cubes of the semimajor axes of proportional to the cubes of the semimajor axes of their orbits.their orbits.