Please help settle this dispute

[ideasrule]

Homework Statement

I'm involved in a dispute about conical pendulums, and my friend, for whatever reason, doesn't want to believe me until somebody backs me up. Assume t is the angle that the string makes with the horizontal.

Homework Equations

F=ma

The Attempt at a Solution

Tsint = mg
Tcost = mw^2*r = mw^2*Lcos(t)
T=mw^2*L, where L is the length of the string

Dividing gives:

sin t = g/(w^2*L)

This implies that if w^2*L is smaller than the acceleration of gravity, a conical pendulum cannot maintain a constant angle t with the horizontal.

P.S. this is actually not a homework question. I'm almost certain my calculations are right, but my friend insists that there's no minimum rotation speed for a conical pendulum.

 

[denverdoc]

Ideas,
I'm not iterested in weighing in one way or another, but the math seems ok by my reckoning--I get tan (x) = g/(w^2*L) but assume that since both approximate to t, you are using that. But if you use that than the angle must be small which implies that sin (t)= tan (t) is small. May be I'm not understanding the last line of the proof.

 

[sylas]

Homework Statement

I'm involved in a dispute about conical pendulums, and my friend, for whatever reason, doesn't want to believe me until somebody backs me up. Assume t is the angle that the string makes with the horizontal.

Homework Equations

F=ma

The Attempt at a Solution

Tsint = mg
Tcost = mw^2*r = mw^2*Lcos(t)
T=mw^2*L, where L is the length of the string

Dividing gives:

sin t = g/(w^2*L)

This implies that if w^2*L is smaller than the acceleration of gravity, a conical pendulum cannot maintain a constant angle t with the horizontal.

You could also say that w^2 cannot be less than g/L. When the angle with the horizontal is close to 90 degrees (that is, when the bob is moving slowly and hanging close to straight down, you have sin t being close to 1 and the angular velocity w of the pendulum approaches the minimum of

$$w = \sqrt{\frac{g}{L}}$$​

This is why pendulums (including conical pendulums) make good time pieces.

Cheers -- sylas

PS. You could also show your friend the wikipedia article on Conical pendulum. It uses the opposite angle convention, measuring the angle from the vertical rather than from the horizontal, but the equation at the end of the article for the period of the pendulum is

$$\textrm{period} = 2 \pi \sqrt{\frac{L \cos \theta}{g}}$$​

which has the same implication; there is a maximum possible period for the conical pendulum.

 

[Dale]

Here is how I did it:
Given a bob of mass m at the end of a string of length L making an angle $\theta$ with the horizontal. The mass is traveling in uniform circular motion in the horizontal plane, calculate the frequency of rotation.

Newton's 2nd law in the vertical direction
$$T \, sin(\theta) - mg = 0$$

Newton's 2nd law in the horizontal direction
$$T \, cos(\theta) = ma$$

Centripetal acceleration
$$a = r\omega^2$$

Radius of circle
$$r=L \, cos(\theta)$$

So we have four equations which we can solve for $\omega$ eliminating a, T, and r. This gives
$$\omega = \sqrt{\frac{g}{L \, sin(\theta)}}$$

So, I agree with your conclusion. To me it was surprising to find that, as the bob hangs almost vertical the frequency does not go to zero. However, I was OK once I realized that the velocity does go to zero.