Calculating Orbit Distance from Equator w/ Particle in Bowl

[Päällikkö]

I suppose I could've used the "Fast particle in a bowl" thread, but as this is a different problem, I decided not to.

If you set a particle to move in a frictionless bowl (radius R) at the velocity v, how far is the orbit from the equator?

I got the following equations:
$$tan\alpha = \frac{v^2}{gr}$$
$$r = Rsin\alpha$$
$$d = Rcos\alpha$$
where g is the constant 9,81$$\frac{m}{s^2}$$ and r the radius of the orbit.

I ended up with $$d = \frac{\frac{v^2}{gR}-\sqrt{(\frac{v^2}{gR})^2+4}}{-2}R$$, which seems a little complicated but gives reasonable answers. I haven't yet figured out why it only works with - in front of the square root (I got both plus and minus when I solved the equations). Is my solution correct?

 

[siddharth]

I got
$$d = \frac {-v^2 + \sqrt {v^4 + 4g^2r^2}}{2g}$$
I think that's the same as what you got.
By putting v=0, I get d=R. Which is a reasonable answer.

I got a quadratic in d. That's where the minus sign came from.

 

[Päällikkö]

I got
$$d = \frac {-v^2 + \sqrt {v^4 + 4g^2r^2}}{2g}$$
I think that's the same as what you got.

Yep, it is the same.

By putting v=0, I get d=R. Which is a reasonable answer.

I got a quadratic in d. That's where the minus sign came from.

I got a quadratic too, from which I get two solutions (the plus/minus-sign). I'm uncertain why the other solution is wrong.

 

[siddharth]

I took the value of d to be from the center of the bowl to the projection of the particle on the vertical axis.

This value must be positive.If I take one of the solutions of the quadratic, then the value of d will always be negative.

Because this is not possible (ie, the value of d must always be positive as the particle moves only in the bowl), I rejected the value and took the other value.

 

[Päällikkö]

This value must be positive.If I take one of the solutions of the quadratic, then the value of d will always be negative.

The outcome cannot be predicted before the final solution(s), or can it?

Is taking a derivative the only "proper" way to figure out which solution is correct?

 

[siddharth]

'd' must be positive because the way I measure it, a negative value of 'd' would mean that the particle is above the hemi-spherical bowl (Where there is no bowl!).Thus, in the context of the question, this answer can be neglected.

 

[Päällikkö]

'd' must be positive because the way I measure it, a negative value of 'd' would mean that the particle is above the hemi-spherical bowl (Where there is no bowl!).Thus, in the context of the question, this answer can be neglected.

Yes, I know it. But without a calculator or taking derivative, is there a way to figure out which of the solutions is right?