How do physicists figure out the equations that model physical phenomena?

Question

For example, we have differential equations that model, for example, resonance, vibration, heat transfer, the motion of a spring, etc. How were these equations first calculated and derived, and to such a degree that we can be sure of their accuracy in modeling real life phenomena? What is the process for actually deriving them like?

In: 4

Answer 1

You do an experiment. Measure the results.

Change one variable. Measure the results. Repeat several times.

Change another variable. Measure the results. Repeat several times.

Now you have a rough idea about how much each parameter affect the results.

For example you apply a known force to a spring and measure its deflection. Apply different force, measure deflection. Repeat several times and you have data relating force to deflection.

Apply the same loads to similar springs with different diameters. Now you have data with diameter as a variable.

Repeat with different materials, different temperatures etc.

If you are interested in how to obtain equations from experimental data, look up numerical analysis.