Acceleration of a magnet as it falls through a metal tube (1 Viewer)

[Cheefi]

How would you draw a graph for the acceleration of a magnet as it falls through a metal tube?
From my understanding:

1. Before entering the tube, it's acceleration would be 9.8
2. As it enters the tube, it's acceleration would become lower
3. As it is traveling through the tube, it's acceleration would go back to 9.8
4. As it exits the tube, it's acceleration would decrease
5. When it is completely out of the tube, it's acceleration would go back to 9.8

However, my teacher said the acceleration as it goes through the tube is not constant, and that it should be a curve. So, what does it actually look like?

 

[InteGrand]

How would you draw a graph for the acceleration of a magnet as it falls through a metal tube?
From my understanding:

1. Before entering the tube, it's acceleration would be 9.8
2. As it enters the tube, it's acceleration would become lower
3. As it is traveling through the tube, it's acceleration would go back to 9.8
4. As it exits the tube, it's acceleration would decrease
5. When it is completely out of the tube, it's acceleration would go back to 9.8

However, my teacher said the acceleration as it goes through the tube is not constant, and that it should be a curve. So, what does it actually look like?

It is indeed a curve.

From Lenz's Law, the differential equation governing the motion of the magnet as it falls through the tube is going to be

, where is the velocity of the magnet (we are taking downwards as the positive direction), is the acceleration, m is the mass of the magnet, g is the acceleration due to gravity, and k is a damping constant which depends on the magnet, pipe material, and pipe geometry. Empirically, that differential equation is what we get.

This can be solved to give velocity as a function of time: . (Assuming v is 0 at t = 0.)

Differentiating with respect to t gives us our acceleration as a function of time: .

So this means acceleration starts off at gravity's acceleration, and attenuates exponentially, becoming 0 in the limit as time goes to infinity.

Eventually of course, your magnet will exit the tube. After this time, the acceleration will quickly get back to g. To calculate exact times, you'll need to calculate k for a given magnet and tube by constructing a simple geometric model of the situation and looking up the conductivity of the metal of the tube.

For the purposes of your a-t graph, it'll look something like this for the period of time that the magnet is inside the tube: http://www.graphsketch.com/?eqn1_co..._lines=1&line_width=4&image_w=850&image_h=525

Further reading: http://arxiv.org/pdf/physics/0702062.pdf, http://en.wikipedia.org/wiki/Magnetic_damping

 

[Cheefi]

Ah okay, so overall, the graph would be more like this?:

 

[InteGrand]

Ah okay, so overall, the graph would be more like this?:

Pretty much, yes.

 

[Cheefi]

So how do you explain the shape of this graph in terms of the induced eddy currents? Do they increase in size and cause a lower acceleration?

Also, is this correct:

1. As the acceleration decreases, the rate of change of flux decreases
2. As a result, the induced emf decreases
3. The eddy currents decrease in size, and eventually cease to exist
4. However, as this happens, nothing is opposing the motion of the magnet, and its acceleration rises back to 9.8
5. The process repeats

 

[InteGrand]

So how do you explain the shape of this graph in terms of the induced eddy currents? Do they increase in size and cause a lower acceleration?

Also, is this correct:

1. As the acceleration decreases, the rate of change of flux decreases
2. As a result, the induced emf decreases
3. The eddy currents decrease in size, and eventually cease to exist
4. However, as this happens, nothing is opposing the motion of the magnet, and its acceleration rises back to 9.8
5. The process repeats

No, the acceleration only comes back to 9.8 once the magnet has come out of the tube.

Its acceleration is always decreasing towards 0 while it is inside the tube (doing so exponentially). In practice, it quickly gets close to 0 and after that its decreasing is very slow, but it is still decreasing.

As the acceleration is decreasing, the speed is actually still increasing, since the acceleration is always positive (though the speed becomes virtually constant pretty quickly.).

When the speed is (practically) at this constant value, the rate of change of flux is constant (not zero), and this causes the magnetic force opposing the magnet's motion to be constant. The magnetic force opposing the motion is equal in magnitude (and opposite in sign) to the weight force of the magnet, hence the two forces cancel out and the net force is 0, hence 0 acceleration.

So basically, as the magnet speeds up inside the tube, the magnetic force keeps increasing and increasing in magnitude (at a slower and slower rate) until it becomes equal in magnitude to the weight force, at which point, since the magnet is no longer speeding up, the magnetic force stays at this weight-force magnitude, causing no more acceleration.

All of this is captured in the differential equation and its solution.

(BTW, the acceleration never actually equals 0, and the magnet never actually reaches a constant speed etc.: it just approaches this constant speed exponentially, like how an exponential graph never actually reaches 0; however, for practical purposes, it gets close enough (like how an e^-x graph gets "close enough" to the x-axis).)