chapter 13 kinetics of particles: energy and momentum methods energy and momentum methods
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CHAPTER 13CHAPTER 13
Kinetics of Particles:Kinetics of Particles:
Energy and Momentum Energy and Momentum MethodsMethods
Method of work and energy – Method of work and energy – Involves relations between displacement, velocity, mass, and Involves relations between displacement, velocity, mass, and
force.force.
Method of impulse and momentum Method of impulse and momentum – –
Involves relations between mass, velocity, force, and time.Involves relations between mass, velocity, force, and time.
Some problems become easier to do.
13.1 INTRODUCTION13.1 INTRODUCTION
When a force is applied to a mass and the When a force is applied to a mass and the mass moves through an incremental distance, mass moves through an incremental distance, the work done by the force is the work done by the force is
13.2 WORK OF A FORCE13.2 WORK OF A FORCE
rdFdU
dzFdyFdxFzyx
ds)(cosFrdFdU
090if0dU 090if0dU 090if0dU
)J(JoulemNorlbftUnits r
rdr
rdF
A
A’
2
121 rdFU
To get the total work done along a path To get the total work done along a path
requires requires
2
1
t
2
1
21dsFds)(cosFU
Notice thatNotice that
When using rectangular coordinatesWhen using rectangular coordinates
2
1
zyx21)dzFdyFdxF(UF
rd
ds
tF
0 s1s 2s
The work is the area under the curve.The work is the area under the curve.
Work of a Constant Force in Rectilinear Work of a Constant Force in Rectilinear MotionMotion
x)(cosFU21
)xx)((cosF12
F
x1
2
)kdzjdyidx(jWdU
Work of the Force of GravityWork of the Force of Gravity
WdydU
2
1
21WdyU )yy(W
21
yWU21
y2
y1
y
dy
A1
A
A2W
Work of the Force Exerted by a SpringWork of the Force Exerted by a Spring
kxF
2
1
21kxdxU )kxkx( 2
1212
221
x1
x2
F
x
Spring undeformed
x
F
F1
F2
x1 x2
x
kxF
))(( 121221 xxxxk
Work of a Gravitational ForceWork of a Gravitational Force
2
1221 dr
r
GMmU
12 rGMm
rGMm
Forces which Forces which do notdo not do work do work (ds(ds = 0 or cos = 0 or cos = 0):= 0):
• weight of a body when its center of weight of a body when its center of gravity moves horizontally.gravity moves horizontally.
• reaction at a roller moving along its track, andreaction at a roller moving along its track, and
• reaction at frictionless surface reaction at frictionless surface when body in contact moves along when body in contact moves along surface,surface,
• reaction at frictionless pin supporting rotating body,reaction at frictionless pin supporting rotating body,
13.3 KINETIC ENERGY OF A PARTICLE. 13.3 KINETIC ENERGY OF A PARTICLE. PRINCIPLE OF WORK AND ENERGYPRINCIPLE OF WORK AND ENERGY
tt maF
If the force doing work is the net force thenIf the force doing work is the net force then
dt
dvm
dt
ds
ds
dvm
ds
dvmv
2
1
t21 dsFU 2
1
2
1
mvdvdsds
dvmv
212
1222
121 mvmvU
221 mvT 1221 TTU 2211 TUT
13.4 APPLICATIONS OF THE PRINCIPLE 13.4 APPLICATIONS OF THE PRINCIPLE OF WORK AND ENERGYOF WORK AND ENERGY
Work the pendulum problem in the text.Work the pendulum problem in the text.
• Wish to determine velocity of Wish to determine velocity of pendulum bob at pendulum bob at AA22. Consider . Consider work & kinetic energy.work & kinetic energy.
glv
vg
WWl
TUT
2
2
10
2
22
2211
• Velocity found without determining expression for Velocity found without determining expression for acceleration and integrating.acceleration and integrating.
• All quantities are scalars and can be added directly.All quantities are scalars and can be added directly.
• Forces which do no work are eliminated from the problem.Forces which do no work are eliminated from the problem.
• Principle of work and energy cannot Principle of work and energy cannot be applied to directly determine the be applied to directly determine the acceleration of the pendulum bob.acceleration of the pendulum bob.
• Calculating the tension in the cord Calculating the tension in the cord requires supplementing the method requires supplementing the method of work and energy with an of work and energy with an application of Newton’s second law.application of Newton’s second law.
• As the bob passes As the bob passes through through AA22 , ,
Wl
gl
g
WWP
l
v
g
WWP
amF nn
32
22
glv 22
13.5 POWER AND EFFICIENCY13.5 POWER AND EFFICIENCY
Power is the rate at which work is done.Power is the rate at which work is done.
dt
dUP
dt
rdF
vF
Units – 1 hp = 550 ftUnits – 1 hp = 550 ft lb/s and 1 Watt = 1 J/slb/s and 1 Watt = 1 J/s
EfficiencyEfficiency
inwork
outwork
inpower
outpower
13.6 POTENTIAL ENERGY13.6 POTENTIAL ENERGY
Close to the EarthClose to the Earth
DefineDefine
2
1
21WdyU )yy(W
21
WyVg
ThenThen
2121 )()( gg VVU
g21VU
Not So Close to the EarthNot So Close to the Earth
DefineDefine
ThenThen
2
1221 dr
r
GMmU
12 rGMm
rGMm
2121 )()( gg VVU
g21VU
rGMmVg r
WR2
For a SpringFor a Spring
DefineDefine
ThenThen
2121 )()( ee VVU
e21VU
221 kxVe
2
1
21kxdxU )kxkx( 2
1212
221
Notice that the work done Notice that the work done by each of these three forcesby each of these three forcesis equal to a change in something that is is equal to a change in something that is a function of position only.a function of position only.The idea of a function of positionThe idea of a function of positionis valid as long asis valid as long asF is conservative. is conservative.It would not work for a force like friction.It would not work for a force like friction.This function, This function, V, is call the potential energy., is call the potential energy.Potential energy is an energy of Potential energy is an energy of position orposition orconfiguration.configuration.
13.7 CONSERVATIVE FORCES13.7 CONSERVATIVE FORCES
Work done by conservative forces isWork done by conservative forces isindependent of the pathindependent of the pathover which work is done.over which work is done.
),,(),,( 22211121 zyxVzyxVU
For shortFor short
2121 VVU
0 rdF
For any conservative forceFor any conservative force
),,( zyxdVdU
An elemental work corresponding An elemental work corresponding to an elemental displacementto an elemental displacement
)dzz,dyy,dxx(V)z,y,x(VdU
),,( zyxdVdU
dz
zVdy
yVdx
xVdzFdyFdxF
zyx
x
VFx
y
VFy
z
VFz
k
zVj
yVi
xVkFjFiFF
zyx
The vector in the parentheses is known as the The vector in the parentheses is known as the gradient of the scalar function gradient of the scalar function V..
VF grad
VF
13.8 CONSERVATION OF ENERGY13.8 CONSERVATION OF ENERGY
VU21
T
1221TTVV
2211VTVT
If the only forces doing work on a system of particles areIf the only forces doing work on a system of particles areconservative, then the total mechanical energy is conserved.conservative, then the total mechanical energy is conserved.
Kinetic plus Potential
Pendulum MotionPendulum Motion
2211VTVT
l
0mvWl0 2
21
gl2v
13.9 MOTION UNDER A CONSERVATIVE 13.9 MOTION UNDER A CONSERVATIVE CENTRAL FORCE. APPLICATION CENTRAL FORCE. APPLICATION
TO SPACE MECHANICSTO SPACE MECHANICS
The gravitational attractive force is conservative.The gravitational attractive force is conservative.So, in space mechanics both energy and angular momentumSo, in space mechanics both energy and angular momentumare conserved since this force is are conserved since this force is both conservative and central.both conservative and central.
O
0v
v
r
0r
0
0P
P sinrmvsinmvr
000
VTVT00
rGMmmvr
GMmmv 2
21
0
2
021
13.10 PRINCIPLE OF IMPULSE AND 13.10 PRINCIPLE OF IMPULSE AND MOMENTUMMOMENTUM
)vm(dt
dF
2
1
t
t12 dtFvmvm
21Imp
For a system of particles For a system of particles external impulses are considered onlyexternal impulses are considered only(remember Newton’s third law)(remember Newton’s third law)
2
t
t1 vmdtFvm2
1
12 vmvm
If no external forces act on the particles thenIf no external forces act on the particles then
13.11 IMPULSE MOTION13.11 IMPULSE MOTION
Non-impulsive forces can be neglected for they are small Non-impulsive forces can be neglected for they are small in comparison (usually) to the impulsive forces. in comparison (usually) to the impulsive forces. If it is not known for sure that the forces are small, If it is not known for sure that the forces are small, then include them.then include them.
Impulsive MotionImpulsive Motion• Force acting on a particle during a Force acting on a particle during a
very short time interval that is large very short time interval that is large enough to cause a significant enough to cause a significant change in momentum is called an change in momentum is called an impulsive forceimpulsive force..• When impulsive forces act on a When impulsive forces act on a particle,particle,
• When a baseball is struck by a bat, When a baseball is struck by a bat, contact occurs over a short time contact occurs over a short time interval but force is large enough to interval but force is large enough to change sense of ball motion.change sense of ball motion.
• Nonimpulsive forcesNonimpulsive forces are forces for are forces for whichwhich
is small and therefore, may is small and therefore, may be neglected.be neglected.
21 vmtFvm
tF
• Impact: Impact: Collision between two bodies Collision between two bodies which occurs during a small time which occurs during a small time interval and during which the bodies interval and during which the bodies exert large forces on each other.exert large forces on each other.
• Line of Impact: Line of Impact: Common normal to the Common normal to the surfaces in contact during impact.surfaces in contact during impact.
• Central Impact: Central Impact: Impact for which the Impact for which the mass centers of the two bodies lie on mass centers of the two bodies lie on the line of impact; otherwise, it is an the line of impact; otherwise, it is an eccentric impact.eccentric impact.
Direct Central Direct Central ImpactImpact
• Direct Impact: Direct Impact: Impact for which the Impact for which the velocities of the two bodies are directed velocities of the two bodies are directed along the line of impact.along the line of impact.
Oblique Central Oblique Central ImpactImpact
• Oblique Impact: Oblique Impact: Impact for which one Impact for which one or both of the bodies move along a line or both of the bodies move along a line other than the line of impact.other than the line of impact.
13.12 IMPACT13.12 IMPACT
• Block constrained to move along Block constrained to move along horizontal surface.horizontal surface.
• Impulses from internal forcesImpulses from internal forcesalong the along the nn axis and from external axis and from external forceforceexerted by horizontal surface and exerted by horizontal surface and directed along the vertical to the directed along the vertical to the surface.surface.
FF
and
extF
• Final velocity of ball unknown in Final velocity of ball unknown in direction and magnitude and direction and magnitude and unknown final block velocity unknown final block velocity magnitude. Three equations magnitude. Three equations required.required.
• Bodies moving in the same Bodies moving in the same straight line, straight line, vvA A >> vvB B ..
• Upon impact the bodies undergo aUpon impact the bodies undergo aperiod of deformation,period of deformation, at the end at the end of which, they are in contact and of which, they are in contact and moving at a common velocity.moving at a common velocity.
• A A period of restitution period of restitution follows follows during which the bodies either during which the bodies either regain their original shape or regain their original shape or remain permanently deformed.remain permanently deformed.
• Wish to determine the final Wish to determine the final velocities of the two bodies. The velocities of the two bodies. The total momentum of the two body total momentum of the two body system is preserved,system is preserved,
BBAABBAAvmvmvmvm
• A second relation between the A second relation between the final velocities is required.final velocities is required.
13.13 DIRECT CENTRAL IMPACT13.13 DIRECT CENTRAL IMPACT
Av
Av
• Period of Period of deformation:deformation:
umPdtvmAAA
• Period of Period of restitution:restitution: AAA
vmRdtum • A similar analysis of particle A similar analysis of particle BB
yieldsyields
B
B
vu
uv e
• Combining the relations leads to the Combining the relations leads to the desired second relation between desired second relation between the final velocities.the final velocities.
BAAB
vvevv
• Perfectly plastic impact, e Perfectly plastic impact, e = = 0: 0: vvv
AB
vmmvmvmBABBAA
• Perfectly elastic impact, ePerfectly elastic impact, e = 1: = 1:
Total energy and total momentum Total energy and total momentum conserved.conserved. BAAB
vvvv
nrestitutio of tcoefficien e
Pdt
Rdt e
uv
vu
A
A
1e0
uv
vu
vu
uv e
A
A
B
B
BA
AB
vv
vv
BA
AB
vv
vv e
• Final velocities are Final velocities are unknown in unknown in magnitude and magnitude and direction. Four direction. Four equations are equations are required.required.• No tangential impulse No tangential impulse
component; tangential component; tangential component of momentum component of momentum for each particle is for each particle is conserved.conserved.
tBtBtAtA
vvvv
• Normal component of total Normal component of total momentum of the two momentum of the two particles is conserved.particles is conserved.
nBBnAAnBBnAA
vmvmvmvm
• Normal components of Normal components of relative velocities before relative velocities before and after impact are related and after impact are related by the coefficient of by the coefficient of restitution.restitution.
nBnAnAnB
vvevv
13.14 OBLIQUE CENTRAL IMPACT13.14 OBLIQUE CENTRAL IMPACT
• Tangential momentum of ball Tangential momentum of ball is conserved.is conserved.
tBtB
vv
• Total horizontal momentum of Total horizontal momentum of block and ball is conserved.block and ball is conserved.
xBBAAxBBAA
vmvmvmvm
• Normal component of relative Normal component of relative velocities of block and ball are velocities of block and ball are related by coefficient of related by coefficient of restitution.restitution.
nBnAnAnB
vvevv
• Note: Validity of last expression does not follow from Note: Validity of last expression does not follow from previous relation for the coefficient of restitution. A similar previous relation for the coefficient of restitution. A similar but separate derivation is required. but separate derivation is required.
• Three methods for the analysis of kinetics Three methods for the analysis of kinetics problems:problems:
- Direct application of Newton’s second lawDirect application of Newton’s second law
- Method of work and energyMethod of work and energy
- Method of impulse and momentumMethod of impulse and momentum
• Select the method best suited for the problem or part of Select the method best suited for the problem or part of a problem under consideration.a problem under consideration.
13.15 PROBLEMS INVOLVING ENERGY 13.15 PROBLEMS INVOLVING ENERGY AND MOMENTUMAND MOMENTUM