Rotating steel wire with bead on it

[mrdoe]

Homework Statement

A small bead with a mass of 100 g slides without friction along a rotating semicircular steel wire, where the semicircle has radius 10cm. The steel wire rotates about a vertical axis at a rate of 2 revolutions per second. Find the positions at which the bead will be motionless relative to the rotating steel wire, if the bead is in the lower half of the semicircle. Gravity points in the direction of the steel wire's axis. All surfaces are frictionless.

2. The attempt at a solution

We have the centripetal force which is in the horizontal direction to the left, which is $\dfrac{mv^2}{r} = 16000\pi^2$. We also have the velocity in the forwards direction, which is not accelerating (?) so there's no force, right? We also have the combined force of gravity and the normal force, which is $100g\sin\theta$, where $\theta$ is the counterclockwise angle from the down direction. But I don't see any force having any component pointing upwards! How does the bead stay motionless!?

 

[kuruman]

You only have two forces, gravity and normal force. The sum of these two, the net force, is directed towards the center that is why is called "centripetal", but it is not a force in itself. Draw a free body of the bead in the lower half of the wire. The normal force has a centripetal component and a vertical component. The bead does not stay motionless! It accelerates.

 

[mrdoe]

However, the only vertical upward force is the vertical component of the normal force, which is there whether the centripetal force is there or not. Then if there were no centripetal acceleration, the bead would fall to the bottom anyways, so it can't stay motionless with respect to the steel wire, right?

 

[mrdoe]

I don't think I've gone into enough detail, so I'll provide a picture, which will help a lot.

This should clear up any misunderstanding.

http://img93.imageshack.us/img93/3598/physicsproblem.png

 

[rl.bhat]

When the wire rotates about its vertical axis, it will be pushing the bead towards the axis horizontally. It is centripetal force Fc. The bead will push the wire with Fc horizontally away from the axis. Resolve this force into two components. One along the radius, and another tangential to the wire.
Similarly resolve the weight of the bead into two components.
Then you can find the condition for the motionless bead.