# Conical Pendulum with free sliding ring

1. Dec 2, 2009

### Identity

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

A particle of mass m is tied to the middle of a light, inextensible string of length 2L. One end of the string is fixed to the top of a smooth vertical pole. The other end is attached to a ring of mass m, which is free to slide up and down the pole. The particle moves in a horizontal circle.
Find the least possible value of $$\omega$$ for the particle to continue in this motion.

2. Relevant equations

$$F_{centripetal} = m\omega ^2 r$$

3. The attempt at a solution

I'm not really sure how to begin with this... I've tried visualising the situation in my head but I can't imagine what would happen in a frictionless situation and how the angular velocity would affect it

thanks

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2. Dec 2, 2009

### kuruman

Begin by finding the tension in each string. To do this draw a free body diagram of the mass that is going around. Write Newton's 2nd law for that mass in the vertical and horizontal direction. You may not assume that the tension is the same in each string. Draw another free body diagram for the ring and write its Newton's Second Law equation. Put the the three equations together to get an expression for the angular speed.

3. Dec 2, 2009

### ehild

Just imagine that the particle moves along a horizontal circle and the ring stays at a certain height, and draw the free-body diagram for both the particle and the ring. The resultant force on the particle is equal to the centripetal force. The resultant force on the ring is equal to zero.

ehild

4. Dec 2, 2009

Thanks :)