# Trouble with ContourPlot

I am trying to graph this implicit equation, but I am unable to do so as the plot produced is an empty grid. May I know how do I fix this problem? Thank you!

g = 9.81;
mu = 0.3;
r = 0.35;
eqn = g/(mu Sin[theta] + Cos[theta]) ==
Sqrt[(mu^2 + 1)/(g^2 + w^4 r^2 (Sin[theta])^2)]
ContourPlot[eqn, {w, 0, 200}, {theta, -10, 10}]

• Indeed an empty plot in view of FindInstance[eqn, {w, theta}, Reals] which results in {}. Jan 9, 2021 at 7:32
• Plot3D[{g/(mu Sin[theta]+Cos[theta]), Sqrt[(mu^2+1)/(g^2+w^4 r^2 (Sin[theta])^2)]},{w,0,200},{theta,-10,10}] and press and drag the mouse to change the orientation of the plot around to see the two expressions
– Bill
Jan 9, 2021 at 7:41
• @Bill hi, thank you so much for the solution, but may I ask why the expression is unable to be plotted in a 2D plot? Jan 9, 2021 at 8:01
• ContourPlot[left==right,...] is only going to plot the level where left==right. If you instead did ContourPlot[left-right,...] then it would show more than one contour and more than one level. If you read reference.wolfram.com/language/ref/ContourPlot.html including clicking on the orange Details and Options and reading what that shows then this might help.
– Bill
Jan 9, 2021 at 8:33

We calculate (LHS-RHS)^2 by (SubtractSides[eqn] // First)^2 and found that its minimum is greater than 0.

g = 9.81;
mu = 0.3;
r = 0.35;
eqn = g/(mu Sin[theta] + Cos[theta]) ==
Sqrt[(mu^2 + 1)/(g^2 + w^4 r^2 (Sin[theta])^2)]
NMinimize[{(SubtractSides[eqn] // First)^2,
0 <= w <= 200, -10 <= theta <= 10}, {w, theta}]


{88.2949, {w -> 200., theta -> 9.71633}}

The minimum 88.2949 means that in the domain  0 <= w <= 200, -10 <= theta <= 10, the equation eqn can't hold.