How to calculate critical speed in circular motion?

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Discussion Overview

The discussion revolves around the calculation of critical speed in circular motion, particularly in a vertical plane. Participants explore different formulas for critical speed and the conditions under which they apply, as well as a specific example involving a rock in a bucket moving in a vertical circle.

Discussion Character

  • Debate/contested
  • Mathematical reasoning
  • Conceptual clarification

Main Points Raised

  • One participant cites two different formulas for critical speed: v = sqrt(g*r) and v = sqrt(2*g*r), questioning the correctness of each.
  • Another participant suggests that the first formula applies when the normal force (n) is zero, while the second applies when the normal force equals the weight (w), indicating different scenarios.
  • There is a call for clarification on the variables used in the equations, particularly what 'w' and 'n' represent.
  • A specific example is presented involving a 500 g rock in a bucket swung in a vertical circle, asking for the minimum speed at the top of the circle to maintain contact with the bucket.
  • Participants discuss the implications of the normal force in relation to maintaining contact with the bucket, questioning what happens when the rock loses contact versus when it stays in contact.

Areas of Agreement / Disagreement

Participants express differing views on which formula for critical speed is correct, with no consensus reached. The discussion remains unresolved regarding the applicability of each formula and the conditions under which they hold true.

Contextual Notes

There is ambiguity regarding the definitions of the variables used in the equations, and the discussion does not clarify the assumptions underlying each formula. The specific conditions of the problem involving the rock in the bucket are also not fully resolved.

Who May Find This Useful

This discussion may be useful for students and educators interested in circular motion dynamics, particularly in understanding the concept of critical speed and the application of Newton's laws in different scenarios.

jayadds
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In one textbook, it says that the critical speed is the minimum speed at which an object can complete the circular motion. It gives the formula:
v = square root of (g*r)
However, in another textbook it says that the formula is:
v = square root of (2*g*r)

How can there be two different types of equation for critical speed? Which one is correct?
It's funny because they both start off with the same Newton's second law: w+n = mv^2/r

For the first equation, they said that critical speed occurs when n = 0 whereas for the second equation, they said that critical speed occurs when n = w. Which one is correct?
 
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I assume we're discussing motion in a circle in a vertical plane.
For future reference, it helps to specify what all your variables are. We can't see the textbook, so have to guess what w and n are.

The centripetal acceleration needed is v^2/r.
For the string to remain tense (is this your 'n'?), gravity must be less than this: g < v^2/r.
The critical speed is therefore sqrt(g*r).
The second textbook might be discussing a different scenario. Or it might simply be wrong.
 
haruspex said:
I assume we're discussing motion in a circle in a vertical plane.
For future reference, it helps to specify what all your variables are. We can't see the textbook, so have to guess what w and n are.

The centripetal acceleration needed is v^2/r.
For the string to remain tense (is this your 'n'?), gravity must be less than this: g < v^2/r.
The critical speed is therefore sqrt(g*r).
The second textbook might be discussing a different scenario. Or it might simply be wrong.

Hi, just let me put it in a better way. How would you solve this question?

You hold a bucket in one hand. In the bucket is a 500 g rock. You swing the bucket so that the rock moves in a vertical circle 2.2 m in diameter. What is the minimum speed the rock must have at the top of the circle if it is to always stay in contact with the bottom of the bucket?

Btw, is this minimum speed regarded as its "critical speed"?

In this example, I know that you would start off with Newton's second law:

N + w = mv^2/r (where N = contact force, w = weight, m = mass, v = velocity and r = radius)
To find the minimum speed, what would be the next step from there?
 
What do you think N will be if it loses contact? If it stays in contact?
 

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