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.