202 |
Week 4: Systems of Particles, Momentum and Collisions |
4.4: Center of Mass Reference Frame
lab frame
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CM frame |
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m1 |
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x |
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vcm |
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xcm |
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m2 |
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Figure 53: The coordinates of the “center of mass reference frame”, a very useful inertial reference frame for solving collisions and understanding rigid rotation.
In the “lab frame” – the frame in which we actually live – we are often in some sense out of the picture as we try to solve physics problems, trying to make sense of the motion of flies buzzing around in a moving car as it zips by us. In the Center of Mass Reference Frame we are literally in the middle of the action, watching the flies in the frame of the moving car, or standing a ground zero for an impending collision. This makes it a very convenient frame for analyzing collisions, rigid rotations around an axis through the center of mass (which we’ll study next week), static equilibrium (in a couple more weeks). At the end of this week, we will also derive a crucial result connecting the kinetic energy of a system of particles in the lab to the kinetic energy of the same system evaluated in the center of mass frame that will help us understand how work or mechanical energy can be transformed without loss into enthalpy (the heating of an object) during a collision or to rotational kinetic energy as an object rolls!
Recall from Week 2 the Galilean transformation between two inertial references frames where the primed one is moving at constant velocity ~vframe compared to the unprimed (lab) reference frame, equation 197.
~xi′ = ~xi − ~vframet |
(400) |
We choose our lab frame so that at time t = 0 the origins of the two frames are the same for simplicity. Then we take the time derivative of this equation, which connects the velocity in the lab frame to the velocity in the moving frame:
~vi′ = ~vi − ~vframe |
(401) |
I always find it handy to have a simple conceptual metaphor for this last equation: The velocity of flies observed within a moving car equals the velocity of the flies as seen by an observer on the ground minus the velocity of the car, or equivalently the velocity seen on the ground is the velocity of the car plus the velocity of the flies measured relative to the car. That helps me get the sign in the transformation correct without having to draw pictures or do actual algebra.
Let’s define the Center of Mass Frame to be the particular frame whose origin is at the center of mass of a collection of particles that have no external force acting on them, so that the total momentum of the system is constant and the velocity of the center of mass of the system is also constant:
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(402) |
P tot = Mtot~vcm = a constant vector |
or (dividing by Mtot and using the definition of the velocity of the center of mass):
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~vcm = |
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mi~vi = a constant vector. |
(403) |
Mtot |
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Week 4: Systems of Particles, Momentum and Collisions |
203 |
Then the following two equations define the Galilean transformation of position and velocity coordinates from the (unprimed) lab frame into the (primed) center of mass frame:
~xi′ = ~xi − ~xcm = ~xi − ~vcmt |
(404) |
~vi′ = ~vi − ~vcm |
(405) |
An enormously useful property of the center of mass reference frame follows from adding up the total momentum in the center of mass frame:
P~ tot′ |
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X mi~vi′ = X mi(~vi − ~vcm) |
ii
XX
= ( mi~vi) − ( mi)~vcm
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= Mtot~vcm − Mtot~vcm = 0 (!) |
(406) |
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The total momentum in the center of mass frame is identically zero! In retrospect, this is obvious. The center of mass is at the origin, at rest, in the center of mass frame by definition, so its velocity
′ ~ ′ ′
~vcm is zero, and therefore it should come as no surprise that P tot = Mtot~vcm = 0.
As noted above, the center of mass frame will be very useful to us both conceptually and computationally. Our first application of the concept will be in analyzing collisions. Let’s get started!