How to calculate critical speed in circular motion?

[jayadds]

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?

 

[haruspex]

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.

 

[jayadds]

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?

 

[haruspex]

What do you think N will be if it loses contact? If it stays in contact?

 

[PJC01]

The concept of critical speed in circular motion can be understood in two different ways, leading to the two different equations. The first equation, v = √(g*r), is based on the idea that critical speed is the minimum speed required to overcome the gravitational force (g) and maintain circular motion at a radius (r). In this case, the centripetal force (mv^2/r) is equal to the gravitational force (mg), hence n = 0. This approach is commonly used in introductory physics courses.

The second equation, v = √(2*g*r), is based on the idea that critical speed is the minimum speed required to overcome both the gravitational force and the centrifugal force (mv^2/r) acting in the opposite direction. In this case, the net force (mv^2/r - mg) is equal to zero, hence n = w. This approach is commonly used in advanced physics courses and is a more accurate representation of the forces involved in circular motion.

Both equations are correct in their respective contexts and can be used depending on the level of understanding and the specific problem being solved. As for which one is "correct," it ultimately depends on the context and the specific definition of critical speed being used. It is important to understand the underlying principles and assumptions behind each equation in order to use them correctly.