speed of conical pendulum at angle alpha to horizontal

1. The problem statement, all variables and given/known data

Assuming we know the length of the string [itex]L[/itex], radius of the swept out circle [itex]r[/itex], angle formed by string and centre of circle, [itex]\theta[/itex], and angle the swept out circle is to the horizontal, [itex]\alpha[/itex], what is the speed, [itex]v[/itex], of the mass if it is constant?

picture: http://ift.tt/1imelD8

2. Relevant equations


3. The attempt at a solution

So in trying to solve this, I broke it down into x and y components and in the horizontal plane, I got:

[itex]\text{F}_{\text{net}} = \text{T}\,\sin(\theta + \alpha)[/itex]

[itex]ma_{x} = m(a\cos \alpha) = \text{T}\, \sin (\theta + \alpha)[/itex]

[itex]\implies \text{T} = \frac{ma\cos \alpha}{\sin(\theta + \alpha)} [/itex]

The angle, [itex]\theta + \alpha[/itex] arrises due to some geometry where if i tilt my axis so that [itex]\text{F}_{g}[/itex] points directly down on the negative y-axis and [itex]\text{T}[/itex] points into the first quadrant, the angle from the positive x-axis is [itex]\frac{\pi}{2}-(\theta + \alpha)[/itex]

and for the vertical plane i get:

[itex]\text{F}_{\text{net}} = \text{T}\cos (\theta + \alpha) – mg[/itex]

[itex]ma_{y} = m(a\sin \alpha) = \text{T} \cos (\alpha + \theta) – mg[/itex]

and substituting [itex]\text{T}[/itex] into our equation I get;

[itex]a\sin \alpha = \frac{a \cos \alpha}{\sin (\theta + \alpha)} \cos (\theta + \alpha) – g[/itex]

since [itex]a = \frac{v^2}{r}[/itex], I substitute and isolate for [itex]v^2[/itex] to get

[itex]v^2 = \frac{gL\sin (\theta) \sin (\theta + \alpha)}{\cos (\theta + 2\alpha)}[/itex]

I am just wondering if I did this properly? I mean if I let [itex]\alpha = 0[/itex], then I end up with

[itex]v = \sin \theta \sqrt{\frac{gL}{\cos \theta}}[/itex], which is what I got when I was working with a normal conical pendulum, however, I feel that as the mass starts to approach the ground then it should speed up. Is it even possible to let [itex]v[/itex] be constant and is my approach correct?


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