Jon P. answered • 06/15/15
Tutor
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Studied honors physics at Harvard, worked with many physics students
I'm not sure that the force between the objects actually matters.
Look at it from the point of view of momentum and the center of mass of the system.
Think of the two particles as being on a line, where 0 is the starting location of m1. That means that m2 is at d. The total moment of inertia around point 0 is m2d, so the center of mass is at m2d / (m1 + m2).
At t0, m1 has momentum m1v, and m2 has 0 momentum, since it is at rest -- for a total momentum of m1v. Regardless of the force between the objects, the total momentum of the system stays constant.
Since the total mass of the system is m1 + m2, the total momentum of the system can also be stated as (m1 + m2)vc, where vc is the velocity of the center of mass. So:
(m1 + m2)vc = m1v
vc = m1v / (m1 + m2)
When the particles collide, you can consider them to be at the same location, since they are small, which means that they are both at the center of mass. So where is the center of mass?
It started at m2d / (m1 + m2) and moved at a velocity of m1v / (m1 + m2). So its location at t1 is just the following:
m2d / (m1 + m2) + (t1 - t0) m1v / (m1 + m2) =
(m2d + (t1 - t0) m1v) / (m1 + m2)
Since m1 starts at 0 and ends up there when the collision occurs, then that's the distance that m1 travels during that time interval -- (m2d + (t1 - t0) m1v) / (m1 + m2)
I think that's correct, but it's worth thinking through my logic and rechecking my algebra.
Jon P.
tutor
I don't have a specific link about this, but I can answer questions...
The momentum of the system is conserved because there is no EXTERNAL force on the system. If there were some other force coming from somewhere else, such as an electric or gravitational field from some other object, then the total momentum of the two particles would not be constant. But all the forces are internal, that is, just the attraction between the two bodies. If you look at the momentum of just one of the masses, it is NOT conserved because the other mass is exerting a force on it. But when you look at the entire system combined, there is no external force, so the total momentum is not changed. It can be EXCHANGED between the two bodies, but the total will remain constant.
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06/16/15
Glen C.
Fascinating. Thank you Jon.
Does it matter where you choose to calculate the moment of inertia? You chose m1's starting point, thus your moment of inertia was m2d. Could I just as correctly chosen m2's starting position instead?
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06/16/15
Glen C.
Jon, your vc does not take into account the fact that m2 (starting at rest) will be accelerating towards m1 (starting at initial velocity v1 presently. Yet your answer is correct. How is this okay?
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06/16/15
Jon P.
tutor
That's a good observation! Technically it doesn't matter in this case, but in other cases it does.
Moment of inertia is always relative to (or "around") a specific point. When you're using it to find the center of mass, it doesn't technically matter which point you choose. But you're best off picking a point that makes the calculations easiest. In this case, either m1's starting point or m2's would be OK, but m1 is better because it's located at 0 on our measurement scale.
In other cases, though, it's important to choose the right point. That's when there is rotation around a particular point or axis (angular momentum) or a rotational force (torque). That's a more common usage of moment of inertia, and in those cases, you must choose the point around which the rotation is occurring.
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06/16/15
Jon P.
tutor
My previous answer was regarding the choice of reference point for moment of inertia. As for the calculation of vc, it's true that m1 is accelerating towards m2, but at the same time m2 is accelerating towards m1. Even though both particles accelerate, the center of mass moves at a constant velocity. That's required by the conservation of momentum. In a multi-particle system, the total momentum is equal to the total mass times the velocity of the center of mass. Since the mass isn't changing, and neither is the total momentum, then the velocity of the center of mass is also constant, regardless of the forces between the two particles.
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06/16/15
Glen C.
Got it, thanks a lot Jon!
-glen
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06/16/15
Jon P.
tutor
You're welcome!
By the way, the problem would have been ENTIRELY different if they had actually asked you to find t1. Since they said you could use t1 and t0 in the answer, essentially as givens, the particular force and its effect on the particles' motions didn't turn out to matter. No matter how long it took the particles to collide, the expression for the location of the collision would be the same. However, if you actually needed to find t1, the specific force between the particles would have made a big difference.
It's easy to see why. If there were no force, then particle 1 would have continued at its initial speed until it collided with particle 2. But if there were a strong force, the particles would have been accelerated together more quickly and the collision would have occurred sooner. So the details of the force would be important in determining the time to collision. It's possible to calculate, but it is much more difficult, requiring the use of calculus and differential equations.
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06/16/15
Glen C.
06/16/15