Stephen A. Lawrence wrote:
Hank Scudder wrote:
Stephen,
I respectfully disagree with your analysis. The rotation of the ball comes from friction with the track as the ball is attracted by the magnet. No friction, the ball would just slide along the track. The magnet doesn't have anything to do with the angular momentum directly.
Right. That's what I was trying to say, I think.
Send the ball out into space (in orbit), and slide the ramp to the right while pressing it against the ball, and the ball will spin without any magnet or gravity present.
Oh, well, this is all about picking nits so I might as well pick another one here. (This is the only way to get an accurate picture of what's going on in a complicated mechanics problem, actually: pick nits until you run out of them.)
The _rotation_ of the ball is due entirely to the friction of the ramp on the ball and the magnet does not directly affect it.
But the angular momentum of the ball depends on what origin you choose to measure it at, and in general it is affected by the magnet.
Remember, angular momentum of a point mass is defined as
L = r x P (unless I have it backwards)
where "r" is the position vector of a point mass, P is its linear momentum (a vector), and "x" is the vector cross product.
Torque is defined as
N = r x F (unless I have it backwards)
where "r" and "x" are as above, and "F" is force (a vector).
If r is not parallel to F, then the torque is nonzero, and the rate of change of the angular momentum is also nonzero, since dL/dt = N.
The angular momentum of a rigid body is the integral over its volume of the angular momenta of all its infinitesimal pieces, viewing them as point masses. This turns out to be the _sum_ of the angular momentum of its rotation (measured in the object's center of mass coordinates) and the angular momentum of the object's center of mass due to its linear motion. This doesn't tend to be highly intuitive.
Now, if we put the origin at "Ball Finish" in
http://physicsinsights.net/images/Plausible-field.png
then a quick glance shows that, when the ball is on the ramp, the force exerted on it by the magnet is not parallel to its position vector, which lies on a line from "Ball Finish" to wherever the ball happens to be. So, the torque exerted on the ball by the magnet is nonzero while the ball is going up the ramp, if we measure it about the point "Ball Finish". Similarly, the torque exerted on the magnet by the ball is nonzero while the ball is climbing the ramp.
However, once the ball starts to fall vertically down from the hole -- and from the magnet -- the forces acting on the ball lie along a vertical line, and the ball's position vector also lies along that vertical line, and so neither the magnetic field nor the gravitational field has any further effect on the ball's angular momentum -- once again, as measured at the point "Ball Finish".
Something to keep in mind is that conservation of linear momentum, which is a very familiar law, is an _assumption_ in Newtonian mechanics. However, conservation of angular momentum, though less familiar, is a _theorem_ -- it follows directly from conservation of linear momentum.
Hank
----- Original Message ----- From: "Stephen A. Lawrence" <[EMAIL PROTECTED]>
To: <[email protected]>
Sent: Tuesday, May 10, 2005 2:30 PM
Subject: Re: SMOT
Terry Blanton wrote:
Here's a quick picture showing some of the most important forces (nb -- those black force arrows on the ramp have somewhat arbitrary directions):From: "Stephen A. Lawrence" <[EMAIL PROTECTED]>
If you don't understand this then you need to brush up on your physics.
Let's talk about the physics. A magnetic gradient pulls the ball up a ramp. Suddenly there's a hole in the ramp and gravity pulls the ball through the hole. The ball is still spinning when it falls. What imparted the angular momentum?
http://physicsinsights.net/images/ball-rolling-up-ramp.png
From the point of view of the ball, as it accelerated up the ramp, the ramp itself applied a tangential force to the surface of the ball which caused it to spin; thence came the angular momentum (in the frame of reference of the ball).
Angular momentum must be measured at a particular point in space. It only makes sense to talk about it with regard to a particular origin. In particular, it's conserved, but that statement only makes sense in a situation where you've chosen one point about which to measure the total value of L. So let's say we measure it at the point the ball lands on when it hits the ground after falling through the hole. From that POV, as the ball moves up the ramp, the ball gains angular momentum both because of its spin and because of the motion of its center of mass along a line which doesn't pass through the point we have (arbitrarily) chosen as the center of our coordinate system. At the same time, the RAMP gains angular momentum which is equal and opposite to the angular momentum of the spin of the ball, as a result of the force the BALL exerts on the ramp as it spins up. Finally, the MAGNET gains angular momentum which is equal and opposite to the angular momentum due to the motion of the ball's center of mass along a line which doesn't pass through the origin.
Now, the ramp is not accelerating in these coordinates, despite the force the ball exerts on it. So, the ramp is also being acted on by other forces (it's attached to the apparatus which is attached to the floor which is attached to the ground) and the L gained by the ramp is actually passed to the environment. Similarly, the magnet doesn't accelerate; its L value is also passed to the environment. But what is "the environment"? It's the Earth itself, which is so massive that we don't normally notice tiny changes in its angular momentum due to things like balls rolling up ramps. In other words, the Earth itself provides an essentially infinite source/sink for L, which is one reason why it's not always apparent that L is really conserved in real-world situations.
