burbery1
(Peter Burbery)
January 3, 2023, 1:36am
1
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)]}}
dontdont9
(Bill Nelson)
January 3, 2023, 8:39am
2
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,
Try:
Solve[m g Sin[\[Theta]] == m v^2/r && 0 <= \[Theta] <= \[Pi], \[Theta], Reals]
(*Simple solution on MMA 13.2.0*)
burbery1
(Peter Burbery)
January 3, 2023, 3:02pm
4
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.
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]]]}}*)