# Calculating the average linear speed of all points in a rotating sphere

• Darius Kaufmann
In summary, the average linear speed of all points in the volume of a sphere rotating on a single axis is apparently dependent on the mass and rotation of the sphere.
Darius Kaufmann
Is there a way to calculate the average linear speed of all points in the volume of a sphere rotating on a single axis? Since points closer to the axis of rotation and the poles move slower than points further out, would the average speed be a simple function of r/2 and pi? It would seem that there should be a significant relationship between that average and energy dynamics on both micro and macro scales. For example, points on the equator of a spinning proton are presumed to be moving at c, according to current theories – What about the gradient of speeds approaching the axis and poles?

easier variation: how about a disk ? Well, kinetic energy from rotation is ##{1\over 2} I\omega^2## with ##I= {1\over 2} M R^2## for a disk.
Kinetic energy of a point mass M moving along at speed ##v## is ##{1\over 2} M v^2##

Equate the energies $${1\over 2} \left ( {1\over 2} M R^2\right) \omega^2 = {1\over 2} M v^2\ \Rightarrow \ v^2 = {1\over 2} \omega^2 R^2\ \Rightarrow \ v = {1\over 2} \sqrt 2\ \omega R$$

Can you now do the 3D sphere by yourself ?

Thank you so much -- However, I'm a musician, not a physicist, so I don't even know what all the symbols mean in the above equations - Can you clarify a little? Thanks again for taking the time to respond!

Darius Kaufmann said:
For example, points on the equator of a spinning proton are presumed to be moving at c, according to current theories

This is nonsense. Why do you think that?

Darius Kaufmann said:
Summary: Is there a way to calculate the average linear speed of all points in the volume of a sphere rotating on a single axis?

Is there a way to calculate the average linear speed of all points in the volume of a sphere rotating on a single axis?
I'm wondering what it is, exactly that you want to know. We need some basic things about motion. The average velocity of a stationary spinning disc (in the reference frame of the Lab) is zero - that is if you use the word "average" to signify the "Mean" - add em all up and divide by the number of points' - and it is going nowhere so the mean is zero. But a spinning object has angular velocity.
There are two quantities that indicate the Motion and 'stopability' of a body. In the past they were confused with one another but eventually they were acknowledged to be two different things. One is the Kinetic Energy and the other is the Momentum. KE involves the square of the velocity and that is always positive. Momentum involves Velocity Vectors which have direction as well as speed.
A stationary, spinning sphere clearly has Energy, despite a mean velocity of zero. @BvU discusses the KE idea, above. But the linear momentum is zero so what about the rotation?
Linear Momentum is just mv but angular momentum depends on the distribution of the mass as well as the total mass. Its Moment of Inertia is the result of adding all the individual particle masses times the (distance from the centre of rotation) squared. So the outer parts have a bigger contribution than the inner parts (flywheels usually have the mass in an outer ring for this reason).

BvU
PeroK said:
This is nonsense. Why do you think that?
Wholeheartedly agree.

Must admit I only read the first half of #1 and (re)acted on that.

Reading the second half feels like hearing a chainsaw go through a violin...

sophiecentaur and berkeman

## 1. What is the formula for calculating the average linear speed of all points in a rotating sphere?

The formula for calculating the average linear speed of all points in a rotating sphere is: v = ωr, where v is the average linear speed, ω is the angular velocity, and r is the radius of the sphere.

## 2. How is the angular velocity of a rotating sphere determined?

The angular velocity of a rotating sphere is determined by dividing the total angle of rotation by the time it takes to complete one rotation. This can be expressed as ω = θ/t, where ω is the angular velocity, θ is the total angle of rotation, and t is the time taken to complete one rotation.

## 3. Can the average linear speed of all points in a rotating sphere be greater than the speed of light?

No, the average linear speed of all points in a rotating sphere cannot be greater than the speed of light. According to the theory of relativity, the speed of light is the maximum speed at which any object can travel in the universe.

## 4. How does the radius of the sphere affect the average linear speed of its points?

The radius of the sphere directly affects the average linear speed of its points. As the radius increases, the average linear speed also increases. This is because the points on the outer edge of the sphere have a greater distance to travel in the same amount of time compared to the points on the inner edge of the sphere.

## 5. Is the average linear speed of all points in a rotating sphere constant?

Yes, the average linear speed of all points in a rotating sphere is constant. This is because the angular velocity and radius of the sphere remain constant as long as there is no external force acting on the sphere. Therefore, the average linear speed of all points will also remain constant.

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