Systems of Particles
1
2.003J/1.053J Dynamics and Control I, Spring 2007
Professor Thomas Peacock
2/20/2007
Lecture 4
Systems of Particles: Angular Momentum and
Work Energy Principle
Systems of Particles
Angular Momentum (continued)
τ ext
B =
d
H + vB × P
dt B
τ ext
B : Total External Torque
H B : Total Angular Momentum
P : Total Linear Momentum
From now on, τ ext
B = τ B.
If τ B = 0 and v B = 0 or if B is the center of mass or if v B � v C then H B =
constant (Conservation of Angular Momentum).
d
You may be familiar with τ B = dt
H B (only valid if v B = 0 or v B � P ).
Angular momentum H B of a collection of particles about point B is given by:
HB =
N
�
hBi
i=1
�
where hBi = r i × mi v i .
If (H B ) is the sum of the angular momenta of the individual particles about
point B,
Cite as: Thomas Peacock and Nicolas Hadjiconstantinou, course materials for 2.003J/1.053J Dynamics and
Control I, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology.
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Systems of Particles
2
Figure 1: Angular momentum about B for a system of particles. Each particle
�
has mass mi positions ri with respect to the origin and ri with respect to B.
�
The center of mass C has positions rc with respect to B and ρi with respect to
each point mass mi . Figure by MIT OCW.
HB
=
n
�
=
n
�
n
�
�
r i × mi v i =
i=1
=
i=1
n
�
�
i=1
�
(r c + ρi ) × mi vi
�
(r c × mi v i ) +
i=1
=
�
r i × mi r˙ i
�
rc × M vc +
n
�
ρi × m i v i
i=1
n
�
ρi × mi v i
i=1
where we have used
Therefore, we write:
�
mi v i = M v c
�
H B = H C + rc × P
Notice that v B does not appear in this equation.
The angular momentum about B is the angular momentum about the center of
mass (C) plus the moment of the system linear momentum (M v C = P ) about B.
We will use these equations for rigid bodies. With rigid bodies will need to use
moments of inertia.
Cite as: Thomas Peacock and Nicolas Hadjiconstantinou, course materials for 2.003J/1.053J Dynamics and
Control I, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology.
Downloaded on [DD Month YYYY].
Systems of Particles
3
Work Energy Principle
W12 =
n �
�
i=1
r2i
F i · dr i =
r1i
n �
�
t2
t1
i=1
d
p · v dt =
dt i i
�
t2
t1
�
�
n
�
d 1
mi (v i · v i ) dt
dt 2
i=1
= T2 − T 1
where:
T =
n
�
1
2
i=1
W12
=
=
n
�
(W12 )i =
i=1
n � r 2i
�
i=1
�n
� r2i
�ni=1 �rr1i2i
i=1 r 1i
mi (v i · v i )
n �
�
i=1
r2i
F int
· dr i +
i
r 1i
Fi · dr i
r1i
n �
�
i=1
r2i
F ext
· dr i
i
r1i
int
F int
· dr i = W12
i
ext
· dr i = W12
F ext
i
int
W12
=
n �
�
t2
n �
�
t2
i=1
=
i=1
=
�
t2
�
t2
t1
int
=
W12
F iint · v i dt
t1
t1
n
�
fij · v i dt
j=1
j�=1
n �
n
�
(f ij · v i + f ji · v j )dt
i=1 j>1
t1
n �
�
f ij · (v i − v j )dt
i=1 j>1
This is non-zero in general.
Cite as: Thomas Peacock and Nicolas Hadjiconstantinou, course materials for 2.003J/1.053J Dynamics and
Control I, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology.
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Systems of Particles
4
Figure 2: Relative velocity probably has a component in the direction of f ij .
The figure shows two random points with randomly chosen velocities. Unless
the difference between the velocities of the two points is zero or perpendicular
to the direction of force f ij , f ij · (v i − v j ) will not be zero; there would be some
component in the direction of f ij . Figure by MIT OCW.
No reason that difference between velocities should not have a component in
the direction of f ij .
If particles are parts of a rigid body system, then there is no relative motion in
the direction of f ij (e.g.)
Figure 3: Two point masses connected by a rod. This is an example of a rigid
body where due to the rod, there is no relative motion of the two point masses
at each end when the rigid body moves. Figure by MIT OCW.
d
|r − r j |2 = 0
dt i
�
�
d
(r i − r j ) · (r i − r j ) = 0
dt
2(r i − r j ) · (v i − v j ) = 0
Internal forces f ij are along the direction (r i − r j ).
f ij · (v i − v j ) = 0.
Therefore, for a rigid body system we have proved:
Cite as: Thomas Peacock and Nicolas Hadjiconstantinou, course materials for 2.003J/1.053J Dynamics and
Control I, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology.
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Examples
5
f ij · (v i − v j ) = 0
int
Therefore, W12
must be 0 (or if you show that internal forces do no work).
Thus,
ext
W12
= T2 − T 1
More generally:
ext
int
W12
+ W12
= T2 − T 1
If all external forces are potential forces or the ones who are external do no work
int
and W12
= 0,
ext
W12 = W12
= V1ext − V2ext
�
V = potential work where V ext = ni=1 Viext . Viext is the external force poten­
tial of particle i.
T1 + V1ext = T2 + V2ext
Examples
Example 1
How does l affect the motion? How does θ affect the motion?
No rotations involved. Probably will not need angular momentum.
Kinematics
Describe the motion (kinematics) without forces
Knowing the location of A is equivalent to knowing the location of the center
of mass of M.
v C = vA
M
m
r A = xı̂ r B = xı̂ + sês = xı̂ + s cos θı̂ + s sin θĵ ṙ A = ẋı̂
ṙ B = ẋı̂ + ṡ cos θı̂ + ṡ sin θĵ r̈A = ẍı̂
r̈B = ẍı̂ + s̈ cos θı̂ + s̈ sin θĵ Note: Generalized coordinates. ı̂ and ês are not ⊥.
Important to define coordinates.
Cite as: Thomas Peacock and Nicolas Hadjiconstantinou, course materials for 2.003J/1.053J Dynamics and
Control I, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology.
Downloaded on [DD Month YYYY].
Examples
6
Figure 4: Block on frictionless surface that moves on frictionless road. The mass
(m) can slide down the incline in a frictionless manner. Mass (M ) is free to
move horizontally without friction. If mass (m) is released from rest at the l
position, find the velocity of mass (M ) at the moment (m) reaches the bottom
of the incline. Figure by MIT OCW.
Figure 5: Diagram of kinematics of block on ramp. Need two sets of coordinates.
M only moves in the x-direction. m only moves in the ês direction. Figure by
MIT OCW.
Cite as: Thomas Peacock and Nicolas Hadjiconstantinou, course materials for 2.003J/1.053J Dynamics and
Control I, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology.
Downloaded on [DD Month YYYY].