Minimum Acceleration for Ball to Reach Top of Vertical Ring

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In summary, the conversation discussed a problem involving a small ball moving on the inner surface of a vertical ring with radius R. The ball reaches its maximum height of R/2 and the question was what minimum acceleration would be required for the ball to reach the top of the ring. The attempt at a solution involved using energy conservation, but it was pointed out that taking into account centripetal acceleration would give the correct answer of a=4g/5.
  • #1
PipelineDream
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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!
 
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  • #2
Welcome to PF!

Hi PipelineDream! Welcome to PF! :smile:

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! :rolleyes:

Use centripetal acceleration ! :wink:
 
  • #3


tiny-tim said:
Hi PipelineDream! Welcome to PF! :smile:

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.


tiny-tim said:
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! :rolleyes:

I'm dubious. Can you explain why this is true and what is wrong with the conservation of energy argument in that instance?
 
  • #4
Hi cepheid! :smile:
cepheid said:
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! :biggrin:
 
  • #5
Hey, I see centripetal acceleration works very well :wink: Thanks very much for your advises, they really helped me!
 

FAQ: Minimum Acceleration for Ball to Reach Top of Vertical Ring

What is the problem about moving ball?

The problem about moving ball refers to the study of the motion of a ball, including its position, velocity, and acceleration, as it moves through space. It involves understanding the forces acting on the ball and how they affect its movement.

Why is studying the movement of a ball important?

Studying the movement of a ball is important in many fields, such as physics, sports, and engineering. It allows us to understand the laws of motion and how objects interact with each other, which has practical applications in designing equipment and predicting the trajectory of objects.

What factors affect the movement of a ball?

The movement of a ball is affected by various factors, including its initial velocity, the force of gravity, air resistance, and any external forces acting on it. The surface on which the ball is rolling or bouncing can also impact its movement.

How can we calculate the movement of a ball?

The movement of a ball can be calculated using mathematical formulas, such as those derived from Newton's laws of motion. These equations take into account the initial conditions of the ball, as well as the forces acting on it, to determine its position, velocity, and acceleration at any given time.

What are some real-world applications of studying the movement of a ball?

Studying the movement of a ball has many practical applications, including in sports, where it can help athletes improve their performance and coaches develop strategies. It also has uses in engineering, such as designing machines that use rolling or bouncing balls, and in physics, where it helps us understand the fundamental laws of motion.

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