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MA4248 Weeks 6-7.
Topics Work and Energy, Single Particle Constraints,Multiple Particle Constraints, The Principle of VirtualWork and d’Alembert’s Principle
1
Work and Energy: the English words originated, viaGermanic and Greek branches, respectively, from theProto Indo-European word Werg about 7 k years ago my homepage under courses/Ussc2001/Energy1.pdf
The work done on a particle that is displaced byin a constant force field
F
equals
FThis work has units of energy.
CONSTANT FORCE FIELDS Let us consider this situation in detail. Let
2
Then hence
Aq
Ap,rqp whererF)p(U
and if
2/)t(r)t(rm))t(p(U particle whose trajectory is
then construct the function
A denoteaffine space and choose a point
RA:U by
F)p(U)p(U
UgradF
)t(rq)t(p
is constantthen
since Newton’s second law implies that
)t(rF2/)t(r)t(rmdtd
F
is the net force on a
CONSTANT FORCE FIELDS and
3)t(rF)t(r))t(p(Ugrad
))t(p(U
t]t)t(r[))]t(p(Ugrad[
t))t(p(U))t)t(r)t(p(U
t))t(p(U))tt(p(U
dtd
0tlim
0tlim
0tlim
CONSERVATIVE FORCE FIELDS
4
The argument on page 20 in Week 1-3 Vufoils showthat this total energy is constant for any conservativeforce field. Let us consider the following converse
2/)t(r)t(rm))t(p(U
There exists a force field F and a function U that arefunctions on A (time independent) such that
is constant for
every particle of mass m that moves with net force F.
0)t(r))t(p(F)t(r))t(p(Ugrad Then
hence UgradF
is conservative.
WORK OVER A SMOOTH CURVE
5
Thomas, p. 1062. The work done by a force (field)
F
over a smooth curve parameterized by a smooth
vector valued function on the interval [a,b] is
bt
at
bt
atdt)t(rFrdFW
If UgradF
is conservative then
r
bt
at))a(p(U))b(p(UdtUW
PATH INDEPENDENCE AND COMPONENT TEST
6
F
Thomas, p. 1072 A vector field is conservative
W depends only on the endpoints if and only if
)a(p and of the curve.
)b(p
Thomas, p. 1074 kPjNiMF
is conservative if and only if
yM
xN
xP
zM
zN
yP ,,
SINGLE PARTICLE CONSTRAINT
7
In Tutorial 3, Prob. 5 you computed the trajectory of a ring sliding down a straight rod by assuming either1. that the total energy is conserved, or 2. that the force of constraint is orthogonal to the rodWhy do these assumptions yield the same trajectory?
Consider a particle having mass m that is constrainedto move along a curve parameterized by a function
of a variable s, called a generalized coordinate. Thismay be the case if the particle consists of a ring thatslides along a rigid wire. We will first assume that thecurve does not move so that it is independent of time
]b,a[s),s(hq)s(c
SINGLE PARTICLE CONSTRAINT
8
Therefore, the trajectory of the particle must equal
where t denotes time and s(t) is a function of time
cadtd FF))t(s(hm
2
2
Newton’s second law implies that
]b,a[)t(s)),t(s(hq)t(p
where aF
cF
is the applied force that would be there
if the physical constraint (wire) was removed, and is the force of constraint defined by this equation
SINGLE PARTICLE CONSTRAINT
9
Define ))t(s(h)t(r
over the curve
]t,t[r 21
cF
2
1
2
1
tt
tta
tt
ttc rdFrmrdF
The work performed by
)t,t(W 21cis
SINGLE PARTICLE CONSTRAINT
10
Then
2
1
2
1
tt
tt
tt
ttdtrrmrdrm
)t(E)t(Errm 1kin2kin
2
1
tt
tt 21
dtd
is the change of kinetic energy over the time interval
SINGLE PARTICLE CONSTRAINT
11
If the applied force is conservative UgradFa
0rdFc
2
1
2
1
tt
tt
tt
tta dtrUgradrdF
and
)t(U)t(UdtU 12
2
1
tt
tt
hence )t,t(W 21c is the change in total energy
It equals zero for all time intervals iff
SINGLE PARTICLE CONSTRAINT
12
In this case, it is very convenient to use arc lengthparameterization of the curve, then
and energy conservation implies that the trajectory isdetermined, up to the initial position, by the first orderdifferential equation
221
21 smrrmEkin
)t(UE)t(sm2
SINGLE PARTICLE CONSTRAINT
13
We now consider the case where the particle isconstrained to move along a curve that is moving with time as in Tutorial 5, Problems 4 and 5.
In this case the force of constraint may perform workon the particle, yet it is reasonable to assume that ateach value of time the force of constraint is orthogonalto the curve described at that time.
Note: the curve at a specific time does not describe theactual trajectory of the particle, this very important factis illustrated in Fig. 2.04 on page 31 of the textbook.
SINGLE PARTICLE CONSTRAINT
14
Therefore, the trajectory of the particle must equal
since we now have a time varying family of curves.
cadtd FF))t(s(hm
2
2
Newton’s second law implies that
)]t(b),t(a[)t(s),t),t(s(hq)t(p
and the orthogonality condition implies that
0s)t(FhFs
)t),t(s(hcc
There are 4 unknowns and 4 equations
SINGLE PARTICLE CONSTRAINT
15
We now consider a particle that is constrained to movealong a (possibly moving) surface. Then the trajectoryis determined by the principle that constraint force attime t is orthogonal to the constraint surface at time t
cadtd FF))t(s(hm
2
2
Parameterize the surface at time t is by a (possibly timevarying) function of generalized coordinates
so that at time t )t),t(q),t(q(hq)t(p 21
21 q,q h
Newton implies
SINGLE PARTICLE CONSTRAINT
16
Note that we now have 5 unknowns, the 2 generalizedcoordinates and 3 components of the constraint force.Newton gives us three equations. We need 2 more. They are provided by the orthogonality principle:
22
21c1
1
21c
c
)t,q,q(hFq
q
)t,q,q(hF
rF0
2,1i,0q
)t,q,q(hF
i
21c
SINGLE PARTICLE CONSTRAINT
17
If the applied force is conservative and if the surface isindependent of time then energy is conserved. Energy conservation is not sufficient to determine the motionsince it provides only 1 additional equation.
If the surface is moving then the forces of constraintmay (and usually do) perform work since
dtt
)t,2q,1q(hrFrdFW cc
dtt
)t,q,q(h 21cF
since 0rFW c
SINGLE PARTICLE CONSTRAINT
18
In the previous discussion of a single particle, we usedgeneralized coordinates. However, we could have
usedrectangular coordinates. If we did then we would have6 unknown variables – 3 coordinates x, y, z for theposition of the particle and 3 coordinates of the forceof constraint. These 6 variables are determined (by thesolution of differential equations) by the constraint equations (1 for a surface and 2 for a curve), Newton’ssecond law (3 equations), and the principle that theforce of constraint is orthogonal to the constraint set (2 for a surface, 1 for a curve)
MULTIPLE PARTICLE CONSTRAINTS
19
For a system with N particles, Newton’s 2nd law gives
N,...,1i,FFFrm coni
appi
netiii
3N equations in 6N variables. We need 3N more!
M=3N-f holonomic constraints, given by equationsM,...,1,0)t;r,...,r,r(G N21
give a total of 6N-f equations, we need f more!
These f equations will be provided by the Principle Of Virtual Work. For 1 particle, this principle says that the force of constraint is orthogonal to the surface(M=1) or curve (M=2) that the particle moves on.
MULTIPLE PARTICLE CONSTRAINTS
20
For multiple particle constraints virtual displacements
are any displacements that satisfy
The principle of virtual work says that the total work done bythe forces of constraint over these displacements equals zero
N1 r,...,r
M,...,1,0rNi
1iG
ir i
0rNi
1iFW i
coni
GENERALIZED COORDINATES
21
The set of virtual displacements form a vector spacehaving dimension f = 3N-M, these are the number of degrees of freedom of the system
If we introduce generalized coordinates then
N,...,1i),q,...,q(rr f1ii f1 q,...,q
N,...,1i,qf
1 q
irri
and we can find a basis for this vector space
GENERALIZED COORDINATES
22
Then the principle of work can be expressed by
This holds for all choices of iff
0qq
rNi
1iF
f
1W
iconi
f,...,1,q
f,...,1,0q
rNi
1iF
iconi
These are the f additional equations that we require.
D’ALEMBERT’S PRINCIPLE
23
Then the principle of work can be expressed by
0q
q
rNi
1i irimFf
1
Ni
1i irimFW
iappi
appi ir
The work done by the applied forces, plus the work done bythe inertial forces , in a virtual displacement is zero
ii rm
EXAMPLES
24
Two particles connected by a light rigid rod (pp.29-30)
The forces of constraint are proportional to
0)rr(FrFrFW 2112211
The constraint
1F
2F
21 FF
a|rr| 21
implies that
0)()( 2121 rrrr
and Newton’s second law implies that
21 rr
hence
EXAMPLES
25
Inclined plane (pp.34)
If the block undergoes a virtual displacement
Inclined plane p. 34
ssinmg down the plane then the applied force, gravity, does work
m
sg
generalized coordinateS is the distance downthe inclined plane
s
and the inertial force (oriented up the plane)
ssm which yields the well-known resultdoes work
sings
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