﻿ The Spinning Ball

# The Spinning Ball

Some people argue that if the Earth was indeed spinning at the speed science tells us, it could not possibly keep all the water (and basically everything else) sticking to the planet, and that we should all be hurled out into space.
The intention of the calculations done on this page is to prove those people wrong.
We will do this by comparing the Earth to a Basketball, and to a Basketball-sized solid piece of Lead.

## The Basics

Let's first take a look at some basic facts, and then give an explanation for the given numbers.

Equatorial
6 378 137.000 0.121 0.121 0.121 m
Equatorial
circumference
40 075 017.000 0.760 0.760 0.760 m
Volume 1 083 210 000 000 000 000 000.000 0.007 0.007 0.007
Density 5 513.585 80.855 11 340.000 804 825 976 624 924 000 000 000 000.000 kg/m³
Mass 5 972 370 000 000 000 000 000 000.000 0.600 84.151 5 972 370 000 000 000 000 000 000.000 kg
Gravitational
acceleration
9.799 0.000 0.000 27 244 885 125 074 200.000 m/s²
What do these numbers mean, and where do they come from?
• Radius, Circumference, and Volume of the objects should be pretty self-explanatory, as they simply describe the size of the (roughly) ball-shaped objects we are working with. But you should note that the volume increases with the cube of the radius.
• Mass divided by Volume gives us the Density of our objects. For the actuall Basketball, this is rather small, as a Basketball's volume is mostly made up of air. Lead, on the other hand, has double the Earth's mean density.
• While a largely air-filled actual Basketball weighs only 0.6 kg, an object of the same shape and size, but made of solid lead, weighs in at over 84 kg.

The Gravitational Acceleration is the acceleration of an object in free fall, and is the steady gain in speed caused exclusively by the force of gravitational attraction.
To calculate it, we need to use the gravitational constant of 6.67430×10-11 N×m²×kg–2
The formula is: (Gravitational constant × Mass) / (Equatorial radius)²
As a result, we get the rather well-known 9.8 m/s² for the Earth, and really small numbers for the Basketball and the ball of lead. This huge difference simply comes from the huge difference in mass of our objects.
And as a side note: Since our calculation of the gravitational acceleration is based the equatorial radius of the Earth, this number will differ slightly if your are not on the equator, or not at sea level.

Looking at these numbers, some things become rather obvious:
• The Earth is big, really big. Especially when compared to a Basketball.
• The Earth is heavy, really heavy. Especially when compared to a Basketball.
• Compared to a Basketball, which is largely made up of air, the Earth is quite dense.
• Because of its large mass, the gravitational pull of the Earth is huge when compared to an actual Basketball.
• Even if we make a Basketball-shaped object out of pure lead, it is "only" 140 times heavier than an actual Basketball, and therefore has 140 times the gravitational pull.

## Considerations and Calculations

Obviously, some of these number are either too large or too small to be handled with ease. So from here on, let's write them in scientific notation.

Equatorial radius 6.378×106 1.210×10-1 1.210×10-1 1.210×10-1 m
Equatorial circumference 4.008×107 7.600×10-1 7.600×10-1 7.600×10-1 m
Volume 1.083×1021 7.421×10-3 7.421×10-3 7.421×10-3
Density 5.514×103 8.085×101 1.134×104 8.048×1026 kg/m³
Mass 5.972×1024 6.000×10-1 8.415×101 5.972×1024 kg
Gravitational acceleration 9.799×100 2.737×10-9 3.839×10-7 2.724×1016 m/s²

Now, before we start comparing the ability to hold water, some general remarks:

• The objects we are comparing obviously need to all be in deep space for the comparison to be valid. Otherwise, if for example the Basketball would be on Earth, it could not have water "stick" to it as that would be pulled toward the Earth, because of Earth's much bigger mass.
• The "Earthball" defined above (a Basketball-sized object with the mass of the Earth) has a density that is over 1 billion times larger than that of a Neutron Star (5.9×1017), or of an atomic nucleus (3×1017)

### Calculation #1: One Revolution per Day

Let's assume our objects are all spinning at the same "speed" (rotation period) as the Earth, spinning around once each day.

• The rotation period of the Earth is 86 400 seconds - that is one revolution every day (24 hours × 60 minutes × 60 seconds)
• Based on the rotation period and equatorial circumference, we can now calculate the rotational speed at the equator (Circumference / Period)
Rotation
period
Rotational
speed
Rotations
per day
Earth 8.640×104 4.638×102 m/s 1
Earthball 8.640×104 8.796×10-6 m/s 1

Please note that 463.8 m/s equals 1 669.8 km/h or 1 037.6 mph, which is the source of the "spinning at 1000 miles per hour" claim that is often made for shock value.

The rotational speed of all the Basketball-sized objects is of course identical, as they have the same diameter and the same rotation period.

Let's calculate the force that is trying to hurl things at the surface ob our objects out into space. This is represented the centrifugal acceleration.
The formula for this is (Rotational speed)² / (Equatorial radius) (speed is important).
Since the centrifugal acceleration and gravitational acceleration directly work against each other, we can also immediately compare them to see if surface matter sticks, or if it goes flying.

Centrifugal
acceleration
Gravitational
acceleration
Difference Factor
Earth 3.373×10-2 9.799×100 -9.765×100 m/s² 290.495
Basketball 6.397×10-10 2.737×10-9 -2.097×10-9 m/s² 4.279
Leadball 6.397×10-10 3.839×10-7 -3.832×10-7 m/s² 600.109
Earthball 6.397×10-10 2.724×1016 -2.724×1016 m/s² 4.259×1025

As you can see, the gravitational acceleration is (far) larger than the centrifugal acceleration for all of our objects. Therefore, at the Earth's rotation period, water (or other stuff) will "stick" to the surface of all of them.
If you want to see it in plain numbers: Earth's gravitational acceleration is 9.799 m/s² and the centrifugal acceleration is only 0.034 m/s², which leads to a factor of 290.495 in favor of gravity.

### Calculation #2: Earth's Rotational Speed

So let's speed things up. In fact, let's assume our Basketball-sized objects are spinning at the same rotational speed (not period) as the Earth.
The rotation period is now calculated by Circumference / (Rotational speed).
The number of rotations per day is calculated by 86 400 s / (Rotation period).

Rotation
period
Rotational
speed
Rotations
per day
Earth 8.640×104 4.638×102 m/s 1
Earthball 1.639×10-3 4.638×102 m/s 5.273×107

Our small objects are spinning fast now. Over 52 million times a day. 610 rotations per second. All that just to match the equatorial rotational speed of the Earth - all because of the huge size difference.
So let's see if things still stick to the surface.

Centrifugal
acceleration
Gravitational
acceleration
Difference Factor
Earth 3.373×10-2 9.799×100 -9.765×100 m/s² 290.495
Basketball 1.779×106 2.737×10-9  1.779×106 m/s² 0.000
Leadball 1.779×106 3.839×10-7  1.779×106 m/s² 0.000
Earthball 1.779×106 2.724×1016 -2.724×1016 m/s² 1.532×1010

For the Earth, of course nothing has changed - we haven't touched it.
But for the "light-weight" balls, the tides have turned. At this rotational speed, and resulting centrifugal acceleration, they can now no longer keep stuff at their surface.
And the different in centrifugal to gravitational acceleration is not trivial, either: The centrifugal acceleration is almost 1.8 million m/s² larger than the gravitational acceleration. So all the stuff will leave fast.

### Calculation #3: Let's reach Equilibrium

What we've seen so far is that at a rather pedestrian rate of one revolution every day, all of our objects can keep hold of things at their surface. But if we spin up the rotation, lighter objects will lose grip and fling things that are on their surface out into space.
Which leaves one quesion open: At which rate do our objects have to spin for the centrifugal and gravitational acceleration to cancel each other out?

Since the gravitational acceleration is dictated only by mass and diameter of the object, we have to calculate the rotational speed, and thereby rotation period, at which the resulting centrifugal acceleration exactly matches that gravitational acceleration.
(Rotational speed) = sqrt((Centrifugal acceleration) × (Equatorial radius))
(Rotation period) = (Equatorial circumference) / (Rotational speed)

Centrifugal &
Gravitational
acceleration
Rotational
speed
Rotation
period
Rotations
per day
Earth 9.799×100 m/s² 7.905×103 m/s 5.069×103 s 17.044
Basketball 2.737×10-9 m/s² 1.820×10-5 m/s 4.177×104 s 2.069
Leadball 3.839×10-7 m/s² 2.155×10-4 m/s 3.527×103 s 24.497
Earthball 2.724×1016 m/s² 5.741×107 m/s 1.324×10-8 s 6.526×1012

As you can see, if the Earth's rotation sped up 17 times, we'd run into a problem.
Meanwhile the much lighter Basketball only needs to rotate more than 2 times a day, and it will not be able to keep anything on its surface.

## Conclusion

Even with the claim of "spinning at 1000 miled per hour", the Earth's rotation is still rather slow. Think about it: Spinning anything at only one revolution per day will not strike anyone as being fast. What makes the rotational speed number look so big is the fact that compared to any spinning object we interact with every day, the Earth is really, really big. It has a large radius, therefore the surface at the end of that radius is moving faster than the surface at the end of a smaller radius when rotating with the same period.
Big mass means big gravitional forces. Although the gravitational pull reduces quickly when moving away from the center of mass, at only one revolution per day it is still enough to overcome the centrifugal force, which is mostly dependend on rotational speed.

Put a Basketball into space, give it a spin of one revolution per day, and water will "stick to the surface".
If someone shows you a YouTube video of someone wetting a Basketball (or any other ball, for that matter), spinning it, and showing the water flinging off in all directions - that is simply because a) the ball is spinning at much more than one revolution per day and b) the ball is inside the Earth's gravitational influence, and atmosphere.