A spinning circle

[Dan]

Ok I have a question about a spinning circle. How can a circle spin round? I know it sound weird but I while on a merry-go-round it hit me that the outside of a circle has a bigger circumference than further in the circle, so wouldn’t the edge of the merry-go-round have to be moving faster than the centre of the merry-go-round because it has a bigger distance to travel than the centre does.

 

To try and put it more simple, if the circumference of a circle is 10cm then further in the circle there would be a circumference of 5cm but for the circle to do a complete rotation in say 60 seconds, the outside of the circle would be going at a speed of 10cm per minute but inside the circle it would only be going 5cm per minute.

 

So basically my question is “does a circle spin at different speeds depending on the distance from the centre of the circle?”

 

I hope that makes sense just tell me if it doesn’t.

[Cap'n Refsmmat]

Yes. However, each point in the circle has the same number of rotations per minute.

 

Go stare at an old record for a while and you'll figure it out

[encipher]

The angular speed is the same regardless of distance from the point of rotation, since angular speed is defined as the following:

 

[math]\omega = \frac{d\theta}{dt}[/math]

 

therefore, it is clear that since every point along the merrygoround covers the same angular distance (theta) then their angular speeds are equivalent.

 

However, the linear speed isn't. Looking at it from a standpoint of how much distance does it actually cover in a single rotation, you can tell that it covers more distance the farther out you go.

 

The relationship between linear and angular speed is the following:

 

[math]V_{trans} = \omega r[/math]

 

where omega is the angular speed and R is the distance from the point of rotation to the point at which you want to calculate the speed of.

[[Tycho?]]

Yes, the outside moves more quickly. I dont see what is wrong with this.

[insane_alien]

well, if all points span at the same tangental velocity i would imagine it would be a bit of a mess.

[EvoN1020v]

Since the merry-go-round is a fixed conjoined matter, the whole thing go at the same angular rotational speed.

 

For instance, I hold 2 separate strings with one longer than the other one and a ball (both same masses) is joined at each end. You hold the strings in the same hand and you spin it around above your head. What happens?

[Edtharan]

Yes, the outside does move at a higher tangential velocity than the inner circle.

 

Ice skaters and dancers exploit this effect. If they start spinning with their arms spread out, their hands have a higher tangential velocity, bringing their arms in towards their body, the arms still have this velocity, but because the circumference of the circle is smaller, the velocity means that the dancer will make more rotations as their arms cover the same distance as when they were spread out.

[encipher]

This 'effect' is conservation of angular momentum, and has little to do with the original question asked regarding the angular velocities of different sections of a merry-go-round.

 

Conservation of angular momentum is as follows:

 

[math]L = I\omega[/math]

 

I is the moment of inertia and omega is the angular velocity. When a skater pulls their arms in, the moment of inertia Decreases, therefore due to conservation if angular momentum, the angular velocity must increase. Hence why they begin to rotate much faster when they pull their arms inwards.

[spikerz66]

yes they move at different speeds but they move diffewrent distances at the same time so they cancel each other out. so

[Sisyphus]

Hey, wheels are crazy things. Think about the wheels on your car. At any given instant, the bottom of the wheel is motionless. It's gripping the road, which is motionless. The top of the wheel, however, is moving forward at twice the speed of your car. The center moves at the same speed as your car. But its one object, and it stays together. Chew on that for a while!