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
Rate = distance/time. (Think miles or km per hour, right? Written km/h.)
If you look at a solid disk and spin it one revolution, a point very near the center is going to travel in a much smaller circle than a point on the outer edge.
If you take both those circles and break them and flatten them out I to straight lines you can measure the linear distance traveled. Both did one revolution, which means they moved for the same amount of time. in the same amount of time.
So if the time is the same, and the distances are different, than the speeds have to be different, right?
In mathematical terms, looking at the S = D/t formula the speed is directly proportional to the distance, and inversely proportional to the time. So if the distance goes up then the speed goes up. If the distance remains the same but the time goes down, then the speed goes up.
To think about that intuitively,if you walk a mile in one hour, you went 1 mph. If you walk twice as far in the same hour, you went 2 mph. Conversely, if you walked a mile in just half an hour, then you must have been walking at a rate of 2 mph. Because if you kept walkingat the same rate for a full hour you would have gone two miles, right?
So, yeah, when you’re closer to the axis of rotation of a solid body your linear speed is slower than if you’re further out because you are traveling less linear distance.
Which is why when you start spinning things you need different terms for rates and you can talk about number of revolutions over time, or rotational speed, or radial speed, etc.
No, it’s why launching space bound rockets from near the equator (i.e. French guinea) is so popular (and why nearly all space launches head east). You gain several hundred mph of orbital speed for free when you do.
This is going to be hard to explain to a 5-year old, but remember vinyl records? No? You only know music as a digital media? Damn.
Well, vinyl records are this big flat “plastic” discs that you put on a record player, and as they turn around, a needle moves through the groves on the record and that plays music. But we don’t need that right now.
What do we need? Well, we need to put a record on a record player, put one lego figurine on the outside edge (A), and one near the inside edge (B). A standard record is 30 cm diameter, so A is at 15 cm of the center, and we’ll place B at 5 cm from the center. Our record is going to turn at 33 1/3 rounds per minute. So if we let the record go for an hour, it will have spun around 33.33*60= 1999.8 times, round up to 2000. So both our figurines have made the same amount of revolutions! But have they traveled the same distance? No!
A, on the outer edge, is traveling around a circumference of π x 30 cm diameter, so 94.25 cm. At the same time, B, on the inner edge 5 cm from the center, is going around a circumference of π x 10 cm diameter, so 31.42 cm.
A travelled at a speed of 1.885 km/h, while B travelled at 0.6248 km/h. A went three times as fast as B!
Now how to transpose this to a ball/the earth? Looking at a record, is as if you’re looking down at the earth from one of the poles. A would be standing somewhere on the equator, while B would be somewhere in Canada, a Scandinavian country, or South-Africa.
Every part of the ball has the same *angular* velocity, but the linear tangential velocity is lower closer to the axis of rotation.
Doesn’t it matter if we talking about a solid? A rubber ball is easy but a multi layered planet with strata gases etc is there some variation.
It depends on what measurement of speed you are using to describe the motion.
Lets take the earth as an example we are all familiar with.
Remember that you can describe, measure or calculate these things in different ways for different purposes.
* **Angular Velocity**
Pretend you draw a line from the equator through the center of the earth to the opposite side also on the equator (Say from [Bogota, Colombia to Surabaya, Lampung, Indonesia](https://www.geodatos.net/en/antipodes/colombia/bogota))
At the surface both Bogota and Surabaya are spinning. One way this spin can be measured is the angle covered in a certain time. This is called [Angular Velocity](https://en.wikipedia.org/wiki/Angular_velocity) and for earth it is close to 360 degrees in 24 hours or 15 degrees / hour or 1 degree of angle every 4 hours. **Every place on that line between Bogota and Surabaya is spinning at the same angular velocity.** This measurement of spin speed is helpful for some calculations.
* **Surface Speed**
The more common measurement of speed is the one an outside observer would see if they were looking down at the earth as it spun underneath them.
The earth has a circumference (distance around the middle) of around 40,000 km (25,000 miles) and it spins once a day. By this measure of spin speed at the surface, a person on the equator is travelling at 1667 km/hour (1042 miles/hour)
To calculate the surface speed at other locations you can use math.
At the North Pole and the South Pole, the surface speed is 0. At the equator the speed is the fastest you will get (100% of this speed). At any location between these you will be slower than the equator, but faster than the poles.
[Look here for more detail](https://www.thoughtco.com/speed-of-the-earth-1435093)
linear velocity = radial velocity × radius.
So yes, in regards to linear velocity, the particles further out on the ball are moving faster.
Interestingly- **not always!**
there’s a way to show this- get a superball, and throw it so it under a coffee table at about 45 degrees… it will bounce off the floor, hit the underside of the table, and come back out on your side. Seems weird as hell.
the ball twists itself up inside when it hits the floor, and then untwists, so it has backspin when it hits the table, so it comes back towards you.
When it’s untwisting, the inside has a different rotational speed to the outside.
Depends. If you’re measuring angular momentum, it’s the same everywhere, along the rotation axis.
If you’re measuring the actual speed in m/s for a given part of the ball, then the further from the axis, the faster it moves.
This is a good problem to exercise mental logic. On a Ferris wheel, how fast would you go if you were in the center vs at the edges while it is rotating the same speed? How much distance would you travel in the same amount of time?
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