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mtwatkin

Conservation of angular momentum

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I have been trying very hard to understand the physics equations needed to make a ridid body physics simulator in 3D. The only equation that I don't understand at this point is the formula for change in angular momentum of two rigid bodies as they collide. The equations that I have are: change in w for objA = Ia^(-1)*(ra x J) change in w for objB = Ib^(-1)*(rb x J) where w = angular velocity, r* = vector from center of mass of obj* to the point of collision, J = the impulse vector, and I*^(-1) is the inverse inertia tensor for obj*. If you use the equation for angular momentum L = Iw then you can see that: change in La = ra x J change in Lb = rb x J since the I and I^(-1) cancel each other out. I don't see how this equation can be correct, since ra != -rb, so ra x J != -rb x J, so the (change in La) != -(change in Lb). Conservation of angular momentum in a system says that (La + Lb) before the collision = (La + Lb) after the collision. This means change in La must equal negative change in Lb. Where have I gone wrong???? Please, ease my frusteration with this complicated topic. (Also keep in mind I have had only one course in basic physics, so I could be completely wrong about all of this)[sad]

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Guest Anonymous Poster
What is conserved is the total angular momentum as seen from one single inertial frame of reference. Since the objects change their linear velocity during the collision you cannot simply add the angular momentum as seen from their respective center of gravity and expect it to be constant.

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Quote:
Original post by mtwatkin
I have been trying very hard to understand the physics equations needed to make a ridid body physics simulator in 3D. The only equation that I don't understand at this point is the formula for change in angular momentum of two rigid bodies as they collide. The equations that I have are:

change in w for objA = Ia^(-1)*(ra x J)
change in w for objB = Ib^(-1)*(rb x J)

where w = angular velocity, r* = vector from center of mass of obj* to the point of collision, J = the impulse vector, and I*^(-1) is the inverse inertia tensor for obj*. If you use the equation for angular momentum

L = Iw

then you can see that:
change in La = ra x J
change in Lb = rb x J

since the I and I^(-1) cancel each other out. I don't see how this equation can be correct, since ra != -rb, so ra x J != -rb x J, so the (change in La) != -(change in Lb). Conservation of angular momentum in a system says that (La + Lb) before the collision = (La + Lb) after the collision. This means change in La must equal negative change in Lb. Where have I gone wrong???? Please, ease my frusteration with this complicated topic. (Also keep in mind I have had only one course in basic physics, so I could be completely wrong about all of this)[sad]

As AP says, velocities of objects change aswell. Total anglar momentum is equal to
1: La + Lb + Ra x Pa + Rb x Pb
where Ra is radius-vector from origin* to center of mass of object a (NOT your ra!) (in other words, position of center of mass of object a) and Pa is linear momentum of object a. Same for object b .

*we can choose origin to be in any place.

by the way - your equations doesn't seems to have correct signs. Let impulse acting on a is Ja and impulse acting on b is Jb. Then Ja=-Jb . We can define that J=Ja so it must be
change in La = DLa = ra x J
change in Lb = DLb = - rb x J

change in Pa = DPa = J
change in Pb = DPb = -J
so change in Ra x Pa = Ra x J
so change in Rb x Pb = - Rb x J

so change of total angular momentum will be:
ra x J - rb x J + Ra x J - Rb x J

Let our impulse is applied in point Q
Then, ra = Q-Ra and rb = Q-Rb
(in words: radius vector from center of mass to point of application of impulse is equal to position of application of impulse minus position of center of mass)

So we finally get
(Q-Ra) x J - (Q-Rb) x J + Ra x J - Rb x J
for change of total angular momentum.
Open brackets
(Q x J - Ra x J) - (Q x J - Rb x J) + Ra x J - Rb x J =
Q x J - Ra x J - Q x J + Rb x J + Ra x J - Rb x J =
regroup same terms together
Q x J - Q x J + Rb x J - Rb x J + Ra x J - Ra x J
which is cleanly zero.

So, total angular momentum is conserved :-)

[Edited by - Dmytry on June 19, 2005 2:52:56 PM]

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