I was holding a rubber band ball in my hand earlier and tossing it up in the air at about eye level. I noticed that I could see the shape of individual rubber bands on the axis of rotation on the outside of the ball but the edges of the ball were blurry. This got me thinking.. is a ball spinning slower near the axis than it is at the outer edge? Is the earth spinning faster at the equator than it is at the poles? If speed is d/t then the math makes sense to a layman like me that the ball would be rotating slower at the center and faster on the edges. Please help.
edit: holy shit. balls are fascinating.
In: 439
Yes, it is. That’s why rotation is measured in things like revolutions-per-second or radial speed. Saying that a ball is spinning at 10 kph, doesn’t make sense unless you’re talking about a specific point on the surface.
For example, the Earth is roughly 25k miles* at the equator and turns once every 24 hours. So the land at the equator is going about 1000 mph, but a foot from the axis of rotation travels about 6 1/4 feet in 1 whole day.
*edit: fixed uom
Yes and no.
From a scalar speed perspective (distance / time), the answer is no. As you surmised, close to the poles (axis of rotation) the amount of distance covered in one revolution is less than at the equator. Therefore the tangential speed is higher farther away from the poles.
But from an angular (rotational) speed perspective, the answer is yes. Each part of the sphere completes one rotation (360⁰ or 2π radians) within the same time, so that will be constant throughout the whole sphere.
Simply put, yes. The angular velocity (rpm) is the same obviously but at different points (equator) the path of one rotation is longer and hence it’s actual speed is higher than at other parts.
Yes you are correct.
If you spin a ball, the points at the poles (axis of rotation) are not moving. This is what we call the north and south poles for the earth.
The equator is the part that moves the fastest.
This has some weird effects on Earth:
– professional high jumpers can jump a tiny bit higher when near the equator, since the centrifugal force from spinning around the earth is pushes them just a little bit more up.
– the same thing goes for the earth itself, which actually bulges out near the equator. Think of how a chef spinning a pizza dough makes it stretch out in a circle; the same thing happens to the equator (only very slightly).
– space misssions are best conducted from near the equator, and always launch toward the east. This helps by adding the speed of the spinning earth to the rocket, making it require less fuel to reach orbital speeds.
Every part of a solid sphere rotates at the same angular velocity. For the Earth, this means that every part of the planet, regardless of latitude, rotates at 15 degrees per hour. However, because lines of latitude are parallel, their circumference gets smaller as you move towards the poles. This means that the distance, in miles, that a degree represents gets smaller as well. So, as you move towards the poles, the linear velocity at a particular latitude gets slower so that the angular velocity remains the same.
Let’s ignore real physics and assume you have a perfectly rigid ball spinning in space, because it’s easier to answer your underlying question that way.
In that case, no part of the ball is spinning faster in an angular sense; that is, every part of the ball is moving the same angle of rotation in the same amount of time (with the exception of the line through the ball which defines the axis of symmetry, which isn’t moving).
However, different parts of the ball are definitely spinning at different speeds in the sense of distance traveled. If you look at the outside of the ball, it has to cover a distance of 2 pi R for every time the ball rotates, where R is the outer radius. But if you look at a piece that’s closer to the axis of rotation of the ball, it needs to travel less distance. And if you look at the point at the top and bottom of the spinning ball, it’s not moving at all. And yes, this happens on Earth. The apparent force of gravity is slightly larger at the North Pole and the South Pole than it is at the equator. Some of this is because the Earth itself bulges around the equator. Some of it is because, the further you are away from the axis of rotation, the more force of gravity has to go towards keeping you on the surface of the Earth rather than pulling you into the Earths surface.
Another way to look at it is at a racetrack. If everyone started off side by side and ran at the exact same speed, the people closer to the middle will finish 1 revolution or 1 lap before the person on the outside. If they were to lock arms and run, the people further out would have to run faster to keep up.
No for velocity (what we ordinarily think of as speed), yes for angular velocity (spinning speed).
You’re using the words a little interchangeably and that is causing your confusion. Spinning refers to the circular motion. Moving refers to traveling a distance. Spinning fast means that it turns all the way around more times that something spinning slower, in the same amount of time. So, no, the ball does not spin slower at different points, it is the same spinning speed all over the ball.
But the pieces the ball is made of (or think of it as locations on and in the ball, if you prefer), do travel through space at different speeds in order to have that same spinning speed. So, if you are straddling the axis of a merry-go-round, your feet move in a small circle in one rotation, while at the edge of the merry-go-round, the handles travel a much longer circle. Since that circle is so much longer but happens in the same time as your much smaller circle in the middle, its speed is clearly higher than your feet are going. So, edge is moving fast, but spinning the same speed.
This is one of those cases where a complete answer is about as easy as the ELI5. More of an ELI15, but I’m sure you’ll follow:
A ball spinning on its axis has what’s called Angular Velocity. We call the Angular Velocity w.
The ball spins about an axis. For any point on (or in) the ball distance d (measured perpendicular from the axis), the speed of that point is simply
2 * pi * w * d
So if you have a 2″ ball (1″ radius), spinning at 100 rpm, a point on the “equator” of the ball will be moving at
2 * 3.14 * 100 rpm * 1 in = 628 in/minute
A point near the top of the axis, 1/8″ from the axis will be spinning
2 * 3.14 * 100 rpm * 1/8″ = 78.5 in/minute
It’s that easy.
The entirety of the ball rotates at the same rate. The ball’s surface moves fastest at its equator, slowest at its poles, just like the Earth–at the equator the Earth is moving at around 1,000 mph or so. At mid latitudes (where most people live) it’s about 800 mph. At the poles it’s zero.
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