Minimum Acceleration for Ball to Reach Top of Vertical Ring

[PipelineDream]

Homework Statement

Small ball moves on the inner surface of the vertical ring with radius R. Moving ball reaches maximum height equal to R/2. What minimum acceleration (in vertical direction) is required (to the system of ring and ball) to make the ball reach the top of the ring?

Homework Equations

All are provided in my solution... there might be another solving methods I haven't tried

The Attempt at a Solution

I've tried to apply energy conservation law in this situation: ball has kinetic energy Ek=m(v^2)/2 in the bottom of the ring. All kinetic energy is converted to potential energy when ball reaches the top position (height equal to R/2). I wrote down energy conservation law: m(v^2)/2=mg(R/2) ---> v^2=gR; When we give vertical acceleration a to the system, acting force is equal to m(g-a), not mg. Value of a must satisfy the condition that ball reaches top of the ring (height 2R). Then I've written down again: m(v^2)/2=2m(g-a)R ---> v^2=4(g-a)R ---> gR=4(g-a)R---> a = 3g/4; However, correct answer provided in my textbook is a=4g/5. I can't understand what's wrong with my solution... I would be very thankful for your help!

 

[tiny-tim]

Welcome to PF!

Hi PipelineDream! Welcome to PF!

It's just a sneakier version of those rollercoaster problems …

you haven't taken into account the fact that if the ball only has enough energy to approach the top at zero speed, it will have fallen into the middle long before it gets there!

Use centripetal acceleration !

 

[cepheid]

Hi PipelineDream! Welcome to PF!

It's just a sneakier version of those rollercoaster problems …

Not exactly. I mean, you don't typically take an entire roller coaster and its track and put it in a gigantic elevator that accelerates upwards at acceleration "a," do you? This is what the problem is saying. If there is enough energy in the system for the ball to oscillate back and forth in the loop up to height R/2 on each side when the whole system is stationary in the Earth's reference frame, then what happens if you put the track + ball in a reference frame that is accelerating upwards? I have to admit that right now I'm not sure.

you haven't taken into account the fact that if the ball only has enough energy to approach the top at zero speed, it will have fallen into the middle long before it gets there!

I'm dubious. Can you explain why this is true and what is wrong with the conservation of energy argument in that instance?

 

[tiny-tim]

Hi cepheid!

I'm dubious. Can you explain why this is true and what is wrong with the conservation of energy argument in that instance?

'cos …

i] the reaction force will be zero well before the top, and

ii] it gives the right answer!

 

[PipelineDream]

Hey, I see centripetal acceleration works very well Thanks very much for your advises, they really helped me!