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I need to find the Kinetic Energy gained as a steel bearing ball approaches a Neodymium magnet. I have the value for the magnetic flux density of the magnet, now I need to find a way to calculate the Kinetic Energy gained by the steel ball as it rolls towards the magnet.

Let's assume that the attraction kicks in at x meters away from the magnet. From that point, the ball accelerates towards the magnet. As it gets closer, the acceleration increases as well.

My first question is, what formulae can I use to find the magnetic attraction between the steel ball and the neodymium magnet of specific dimensions. If it requires more variables than an average student can find out, providing a point of reference or even an online calculator would be greatly appreciated. (ANSWERED)

My second question, how can I find the velocity with which the bearing ball strikes the magnet. As I stated, the acceleration constantly increases as the magnetic forces increase. This leads me to think that there is some calculus involved with the calculations, which isn't a problem. Providing a formula for this or a method to follow would be, again, greatly appreciated.

Thanks in advance.

EDIT: I found a great website for calculating the force a neodymium magnet of specific dimensions applies to a steel surface (https://www.kjmagnetics.com/calculator.asp). I also read about position dependent acceleration, which is the case in my experiment. Now I need to combine these to come up with a final equation.

In a video I watched on position dependent acceleration, this formula is used

\begin{align} a(x) &= \frac{-9.81}{(1+(x/6371))^2} \\\\ \end{align}

This example is based on gravity, but I don't see a reason why it wouldn't work with magnetism. Here, 9.81 is the gravitational acceleration of Earth on its surface, and 6371 is the radius of Earth in kilometers.

If I were to apply this equation to the magnet, I would use the acceleration I derived from the link above as in place of 9.81. Please correct me if I'm wrong, but from my knowledge, the magnet doesn't necessarily has a "center of magnetism", meaning that I don't have to necessarily take the ratio of the distance and the radius (x/6371).

So, I found the answer for my first question, but the second question persists. Any help would be appreciated.

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I would hook the bearing to a spring with a known spring constant and use that to find the force applied on the bearing. $F = kx$ is the spring force equation, where X is the distance extended, and k is the spring constant

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