How Does Speed Influence Complete Revolutions in Circular Motion?

[teme92]

Homework Statement

A smooth circular wire of radius a is fixed with its plane vertical. A small ring
threaded on the wire is projected with speed u from the lowest point of the
circle. Taking gravitational acceleration to be the constant g, calculate the
potential energy and the kinetic energy of the ring. Assuming conservation
of energy, show that the ring will describe complete revolutions if:

u^2 > 4ga

Homework Equations

I know all relevant circular motion and SHM equations but don't know where to begin.

The Attempt at a Solution

I genuinely have no idea how to approach this problem. Any help will be much appreciated.

 

[nasu]

Write the potential and kinetic energies of the ring, at the bottom of the wire and top of the wire.
And use conservation of energy.

 

[teme92]

Hi nasu thanks for the speedy reply.

I'm having trouble visualizing the problem. Would the potential energy be equal to mgh + mg(0) as at the bottom of the wire h=0? And for kinetic energy do I use 0.5(m)(v^2)?

 

[nasu]

You don't add the potential energies.
The potential energy at the bottom may be zero, yes, if we measure it from that level.
At the PE at the top point will be mgh, where h is the height of the top pf the circle.

And yes, this is the formula for KE.

 

[teme92]

Ok and for the second part of the question where I'm asked to show that the ring will describe complete revolutions. What would show it describes complete revolutions?

Thanks again for the help.

 

[nasu]

Conservation of energy. I told you already.
But first you need the correct expressions for PE and KE energy.

 

[teme92]

Thanks for your patience, I'm new to these type of problems and I'm finding them tricky to understand.

So conservation of energy is PE=KE

PE=mgh

KE= 0.5(m)(v^2)

mgh=0.5(m)(v^2)

Putting in the form the question requires and I get:

u^2=2gh,

which isn't the required answer. Clearly the 'h' isn't part of the answer so how to I go about getting rid of it?

 

[nasu]

What is h in terms of a, the radius of the circle?

 

[teme92]

I completely forgot 'a' was the radius, I must have been half asleep last night doing this. I was thinking it was acceleration. Thanks a million, I have the solution now.