Thursday 2 December 2010

Conical Pendulum

Describe quantitatively when the conical pendulum spins faster; what's the angle θ between the string and the vertical tube?
-          L: length of the string
-          r: radius of circular motion
-          T = Mg, tension produced
-          mg: mass of the ball on the string
By observation it's obvious that θ tends to 90 degrees.
Approach 1:
(1): Tcosθ = mv2/r
(2): Tsinθ = mg
(2)/(1): tanθ = v2/rg = v2/Lgsinθ
tanθsinθ = v2/Lg
When θ is between 0 and 90 degrees, tanθ and sinθ has the same behavior, L, g being fixed, then when v increases, θ must increase (and tends to 90 degrees)
Approach 2:
Tcosθ = mrω2
When v increases, ω also increase. Then cosθ will increase.
Now we'll find a dilemma:
By Tsinθ = Mgsinθ = mg, we can see when sinθ increase, Mg and mg is still fixed, which leads to a unbalanced equation.

Can you solve the dilemma?

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