Answer 1:
The mathematics is not simple, so if you have bismuth and a magnet, I recommend directly measuring it instead. Put a chunk of bismuth on the front of a cart with low rolling resistance (Legos work nicely) on a flat plane (cookie sheet?) and push it uphill with a magnet. Raise the angle of the inclined plane until the magnet is just unable to push the cart uphill. Measure the angle you had to raise the inclined plane. This gives you the pushing force. Start with Newton's law: F=m*a where m is the total mass of the bismuth+cart. Since gravity is our accelerating force: F=m*g where g=9.81 m/s^{2} on the surface of Earth. But this formula needs one more major correction. Gravity is pulling downward (vertical), but the force between the bismuth and the magnet is only horizontal, and only partly vertical. How much gravitational force is along the incline? From basic trigonometry (which you may have seen already), the *fraction* of force is the sine of the angle from horizontal: F (along the direction plane) = F (vertical) * sin (angle of plane from horizontal)
For example, the sine of 5 degrees is .0872. So a 100g cart on a slope of 5 degrees experiences F = 0.1kg * sin (5 degrees) * 9.81 m/s^{2} = 0.1 * 0.0872 * 9.81 (kg*m/s^{2}) = 0.085 Newton of force down the hill. (1 Newton = 1 kg*m/s^{2}) So if your bismuthmagnet combination is able to just barely repel a 100g cart at an angle of 5 degrees, you know the repulsive force is 0.085 Newton You can thus experiment with different combinations of magnets, and the size & shape of bismuth, to see which gives the strongest repulsion. Once you find the configuration which provides the maximum amount of force, you can calculate how much stronger your magnets would need to be to levitate the bismuth. (Levitation would be when the angle=90 degrees, so sin (90 degrees) =1.) (A caution when calculating sines: calculators and spreadsheets like AppleWorks or Excel sometimes use radians instead of degrees. So try calculating sin (90) and see if you get 1.00. If you get 0.893 instead, then the calculator or spreadsheet is using radians. Fortunately, the conversion is simple: radians = degrees * pi/180, or approximately degrees/57.) Galileo did a lot of work measuring forces and accelerations with an inclined plane. You'll be following in the footsteps of a master!
