how to express solution only in terms of some variables in sympy.solve?

Question

How to express this solution in terms of r, g, and phi?

import sympy as sp
m, N, phi, v, g, r = sp.symbols('m N phi v g r')
sp.solve((-N*sp.sin(phi)+m*v**2/r,N*sp.cos(phi)-m*g),v, dict=True)

yields: $\left[ \left\{ v : - \sqrt{\frac{N r \sin{\left(\phi \right)}}{m}}\right\}, \ \left\{ v : \sqrt{\frac{N r \sin{\left(\phi \right)}}{m}}\right\}\right]$

The result I want is $v:\sqrt{rgtan(\phi)}$

as derived in: https://ocw.mit.edu/courses/8-01sc-classical-mechanics-fall-2016/pages/week-3-circular-motion/11-2-worked-example-car-on-a-banked-turn/

Answer 1

You can add the variable you want to eliminate to the list of variables you want to solve for.

>>> sp.solve((-N*sp.sin(phi)+m*v**2/r, N*sp.cos(phi)-m*g), v, N, dict=True)

[{N: g*m/cos(phi), v: -sqrt(g*r*tan(phi))}, {N: g*m/cos(phi), v: sqrt(g*r*tan(phi))}]