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Solution for equation that models a car on a banked curve?

I have an equation that models a car on a banked curve that I want to solve for the angle:

eqn = m g Sin[\[Theta]] == m v^2/r

For some reason, Solve won't find the solution when I constrain the solution between 0 and pi. Neither of the following lines worked:

Solve[eqn, \[Theta], Assumptions -> {0 <= \[Theta] <= \[Pi]}]
Solve[eqn && 0 <= \[Theta] <= \[Pi], \[Theta]]
Solve[m g Sin[\[Theta]] == m v^2/r && 
  0 <= \[Theta] <= \[Pi], \[Theta]]

How can I solve this equation?
After typing Quit, Solve worked

Quiet@Solve[
  m g Sin[\[Theta]] == m v^2/r && 0 <= \[Theta] <= \[Pi], \[Theta], 
  Reals, Assumptions -> {0 <= \[Theta] < \[Pi]/2, 0 < v < Sqrt[g r], 
    g > 0, r > 0, v < Sqrt[ g r]}]

The solution returned was

{{\[Theta] -> ArcSin[v^2/(g r)]}}
POSTED BY: Peter Burbery
4 Replies

Try:

Solve[m g Sin[\[Theta]] == m v^2/r && 0 <= \[Theta] <= \[Pi], \[Theta], Reals]

(*Simple solution on MMA 13.2.0*)
POSTED BY: Mariusz Iwaniuk

I tried your solution, and it returned the input without solving it. This is the output I obtained

Solve[0.72934584563 g m == (m v^2)/r, 0.817365299581, Reals]

I entered Quit and it worked now.

POSTED BY: Peter Burbery

Try with fresh Kernel ?

$Version
(*"13.2.0 for Microsoft Windows (64-bit) (November 18, 2022)"*)

Remove["`*"];ClearAll["`*"];
Solve[m g Sin[\[Theta]] == m v^2/r && 0 <= \[Theta] <= \[Pi], \[Theta], Reals]

(*{{\[Theta] -> ConditionalExpression[\[Pi] - ArcSin[v^2/(g r)], Or[
And[
Inequality[-(g r)^Rational[1, 2], LessEqual, v, 
       LessEqual, (g r)^Rational[1, 2]], g > 0, r > 0], 
And[
Inequality[-(g r)^Rational[1, 2], LessEqual, v, 
       LessEqual, (g r)^Rational[1, 2]], g < 0, 
      r < 0]]]}, {\[Theta] -> 
   ConditionalExpression[ArcSin[v^2/(g r)], Or[
And[
Inequality[-(g r)^Rational[1, 2], LessEqual, v, 
       LessEqual, (g r)^Rational[1, 2]], g > 0, r > 0], 
And[
Inequality[-(g r)^Rational[1, 2], LessEqual, v, 
       LessEqual, (g r)^Rational[1, 2]], g < 0, r < 0]]]}}*)
POSTED BY: Mariusz Iwaniuk
Posted 1 year ago

Try

Reduce[m g Sin[θ]==m v^2/r&&0<=θ<=\[Pi]&&Element[m|g|r|v,Reals],θ]//Simplify

The result would be simpler if you tell it that m,g,r are positive,

POSTED BY: Bill Nelson
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