Quote by tahayassen
Why must the coils of the spring not touch? How come the position function is a perfect sine wave with gravity? It doesn't make sense to me. As the mass is moving up while below the yaxis, wouldn't it be slowed down by gravity? And why the mass is moving down while above the yaxis, wouldn't it be sped up by gravity?
Does the solution to the equation of motion of a spring explicitly give the sine function as the only solution? So it would be impossible to model the wave if trigonometry was not studied and the sine function was not discovered?

If the coils touch other, the motion will be interfered with and it will no longer be SHM.
In gravity, the same force (the weight) is always in the same direction and doesn't affect the dynamic motion.
The sine function is just a way of describing a bit of geometry (at its simplest). You can relate SHM to circular motion in simple terms, as with the motion of a piston resulting from a crank going round, but, as a general principle, you can't do advanced Physics without bringing in Maths functions  the problems just can't be stated in a way that can be solved if you don't use some form of Maths. Newton, for instance, had to make up his own form of calculus before he could get anywhere.
The sine function is
the solution to an equation  there isn't 'an easier one'.