[syhprum (OP)]

During a discussion with the children about green energy the question arose as to how much energy was stored  in the rotation of the Earth and how long we could go on taping it with tidal power plants so I promised I would calculate.
It soon became apparent that it was beyond my mathematical skills so I resorted to google! ,to calculate the energy stored in a thin rimmed fly wheel is easy and I could just manage a thin disk but after that I am stumped I think I need calculus but I never learned none so could some mathematically skilled person please explain.

[Halc]

During a discussion with the children about green energy the question arose as to how much energy was stored  in the rotation of the Earth and how long we could go on taping it with tidal power plants so I promised I would calculate.
It soon became apparent that it was beyond my mathematical skills so I resorted to google! ,to calculate the energy stored in a thin rimmed fly wheel is easy and I could just manage a thin disk but after that I am stumped I think I need calculus but I never learned none so could some mathematically skilled person please explain.

It is complicated and requires significant integration.  First you need a density curve of Earth, which would be a function in spherical coordinates (how dense matter is at each radius from the center).  You integrate that to get a cylindrical curve (a function of how much mass exists exists at each radius from the polar axis), and then integrate that again to get its angular moment, from which its energy and momentum can be computed given a 23.93 hour day or .0000729 rad/sec.  The rotation is not 24 hours remember.

Fortunately you can skip the hard part and just look it up.  Earth has an angular moment of inertia of about 8e37 kg m².
Plug the .0000729 rad/s angular speed into that and you get 2.14e29 joules, some of which bleeds away every day due to tidal energy.  Yes, they have tidal energy generation plants that utilize this rather than just let it go to waste.

[syhprum (OP)]

I do not really want to do the calculation for the Earth google have already done it for me I just want to improve my maths skills so that I can calculate the energy stored in a rotating homogeguous rotating sphere of density 1,radius 1M rotating at 1 radian per second .
I can find the formula quoted but it goes on about moments of inertia that defeats me.
I can calculate the energy of a disk in a crude way imagining it series of thin edged flywheels of various diameter and adding I could go further and imaging the sphere as a collection of such disks of varying diameter and adding but I want to know the elegant formal way of doing it

Perhaps I could ask the Mathematica program which I have ?.   

[chiralSPO]

If you know the equation for a thin flywheel (as a function of r), then you can integrate that with respect to r from 0 to R to find the energy stored in a disk of radius R, and then you can integrate those to make up a sphere of disk slices (integrating along the axis or rotation will be easiest)

This will end up being a double integral because the equation for the flywheel (ring) has already accounted for one of the dimensions. If you want to start from scratch, it will be a triple integral, where you start with the kinetic energy of each point within the sphere.

[syhprum (OP)]

I left school at 15, if I had stayed until 16 I would have learnt about Integrating and whatnot can you please show the answer to the simple example I quoted.
 

[chiralSPO]

I hope to have some free time over the next few days. If no one beats me to it, I can try my hand at a solution.

[Bored chemist]

This
https://en.wikipedia.org/wiki/List_of_moments_of_inertia
will tell you if you get the right answer.

For additional masochism, you can do some more integration.
You can integrate the kinetic energy of each particle that the sphere is made from.

[Halc]

This
https://en.wikipedia.org/wiki/List_of_moments_of_inertia
will tell you if you get the right answer.

Moment of inertia for a sphere is 2/5 Mr².  Yes, it is masochistic to derive that.
That formula gives the wrong answer (too high) for Earth since it is not uniform density, and the OP talked about computing that for school children.

I do not really want to do the calculation for the Earth google have already done it for me I just want to improve my maths skills so that I can calculate the energy stored in a rotating homogeguous rotating sphere of density 1,radius 1M rotating at 1 radian per second .

Getting angular momentum and energy is just a matter of multiplying that moment by the angular speed once or twice respectively.  The example above has 1 for all variables, so the answer for everything (moment of inertia, angular momentum, and energy) is 0.4, but in different units.

[Bored chemist]

and the OP talked about computing that for school children.

Computing the moment of inertia for school children requires specifying the axis about which you rotate them (and, usually,  whether or not they stick their arms out).

[evan_au]

how long we could go on tapping it with tidal power plants

Tidal power is a very inefficient way of utilising the Earth's angular momentum.

(Fortunately), most of the other ways of changing the Earth's angular momentum are also quite inefficient
- or extremely destructive, like the collision that is thought to have produced the Moon

There will continue to be some tidal power until the Earth's rotation becomes tidally locked to the Moon
- In the meantime, the Moon is slowly receding from the Earth, reducing the Moons contribution to tidal power.
- The Sun's contribution will continue beyond that..
- So we can expect some continuing tidal power for a few billion years - until the Sun's increasing luminosity boils the oceans...

See: https://en.wikipedia.org/wiki/Future_of_Earth#Obliquity

[Halc]

Tidal power is a very inefficient way of utilising the Earth's angular momentum.

(Fortunately), most of the other ways of changing the Earth's angular momentum are also quite inefficient
- or extremely destructive, like the collision that is thought to have produced the Moon

So I posed this question at the dinner table, looking for stupid unlimited-resources ways to engineer power generation from Earth's spin, and the winning answer was to build a sort of inverted railroad track around the equator with the rails sitting on fixed wheels below, each attached to generators that power various parts of the globe.  This gives you a movable ring sitting on roller bearings.  Then you take a sort of carbon nanotube rope sort of like what they'll use to make a space elevator, except secure one end to the near face of the moon and the other end to the moving ring.  Free and almost unlimited power.

Couple of side notes:  The rope will pull up to 23 degrees to the side twice a month.  The bearings need to be able to take that sideways force, sort of like the keel of a sailboat.
Secondly, the more electricity you generate, the greater the tension in the line, and while turning on the device for the first time will increase the force accelerating the moon inwards, continued large scale operation will quickly increase the orbital radius meaning periodically the length of the line needs to be lengthened.  Oh, and the length of the day gets longer at a quicker pace.  More time to get stuff done, but fewer days in the year.
Thirdly, the orbit is a bit eccentric.  We fix that by varying the power generation (the load on the line) until we circularize the orbit.  This is a real problem since the variance is ~45000 km, more than the diameter of Earth.  We need to circularize it first.  There will need to be a way to reel out and in the line at the ring attachment point.  We needed that anyway to solve point 2, but reeling-in requires a fair portion of the energy generated by our operation.

Now to figure out how to generate power from the kinetic energy of the moon, especially since the above idea is giving it more.