Conical Pendulum

Describe quantitatively when the conical pendulum spins faster; what's the angle θ between the string and the vertical tube?

Quantity:

-          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?