Rotations and revolutions

[mfgoode]

I have been working on rotations and revolutions. Kindly clarify.
How many rotations will a penny make if it moves around a fixed similar penny once? How many revolutions?
If we generalise this with different diameters for the fixed disc and moving one could we evolve a golden rule for the number of rotations and revolutions?
We take it that rotation is motion around a centre or axis.
Revolutions means orbiting another body
Mally

[ewmayer]

Originally Posted by mfgoode

I have been working on rotations and revolutions. Kindly clarify.
How many rotations will a penny make if it moves around a fixed similar penny once? How many revolutions?
If we generalise this with different diameters for the fixed disc and moving one could we evolve a golden rule for the number of rotations and revolutions?
We take it that rotation is motion around a centre or axis.
Revolutions means orbiting another body
Mally

For discs of equal size, the outer (moving disc) will make two revolutions (means the same as rotations, the only question in either case is "with respect to what?") about its own axis as it makes one about the boundary of the fixed disc. Both discs have the same radius and hence circumference, so as it moves about the fixed disc, each point on the edge of the moving disc touches a corresponding point on the edge of the fixed disc precisely once. So when the moving disc has traversed 180 degrees around the fixed one it will have only done a half-rotation with respect to the boundary of the fixed one, but because the boundary of the fixed one is also a circle, that 180-degree point of contact is now on the opposite side of where the moving disc started, i.e. its effective rotation with respect to the frame of reference in which the inner disc is fixed is doubled. You can generalize this to discs of arbitrary relative diameter - consider that your homework exercise.

[mfgoode]

Originally Posted by ewmayer

For discs of equal size, the outer (moving disc) will make two revolutions (means the same as rotations, the only question in either case is "with respect to what?") about its own axis as it makes one about the boundary of the fixed disc. Both discs have the same radius and hence circumference, so as it moves about the fixed disc, each point on the edge of the moving disc touches a corresponding point on the edge of the fixed disc precisely once. So when the moving disc has traversed 180 degrees around the fixed one it will have only done a half-rotation with respect to the boundary of the fixed one, but because the boundary of the fixed one is also a circle, that 180-degree point of contact is now on the opposite side of where the moving disc started, i.e. its effective rotation with respect to the frame of reference in which the inner disc is fixed is doubled. You can generalize this to discs of arbitrary relative diameter - consider that your homework exercise.

Dear ewmayer,
Our friend Michael has made the same mistake.
I have been explicit that a rotation is movement around its own central axis
A revolution is movement or orbit around another object.
Here is a copy to my reply to Michael.
Re: rotations and revolutions

--------------------------------------------------------------------------------

Quote:

mfgoode wrote on 20 Jan 04 11:49:
dear Michael,
Im afraid you are off the mark.
Please do the actual experiment and mark the first points of contact and then move the free coin around the fixed one.
You will find that at the half way mark it has already rotated once
Hence on the other half it completes one more.
So the no. of rotations is two around its own axis
No. of revolutions one around the fixed penny. So in ALL 3 rounds
Mally.
On this reasoning the moon always shows one face to the earth
In one revolution By its very motion it faces the earth all the time.
Try the experiment with going round a table or chair but always facing it. You will make one rotation automatically.
Mally



lol, i should've known it was a trick question, it seemed too easy...

-michael

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[mfgoode]

Rotations and Revolutions-2
Regards the number of rotations a penny makes without slipping in revolving around a similar fixed penny, I got answers not too enlightening . In case the members considered my problem too trivial and too obvious to comment upon I will set the record straight.
The moving penny makes 2 rotations with respect to its own centre and one revolution around the fixed penny’s edge equal to its circumference. Hence in all three rounds.
The truth is that for every degree of arc along which it rolls it rotates 2 degrees.
For other unequal diameters simply calculate the length of the path in degrees, multiply by 2 and you have the number of degrees of rotation.
For a fixed disc twice the diameter of the moving disc the number of rotations are four and one revolution . In all five rounds Ref: ‘Mathematical Carnival’ pages 16 and 17. Martin Gardner.
mfgoode

[mfgoode]

Mind Boggling Number.
The largest number that can be written using only 3 digits is 9^9^9.
Mathematician and editor Joseph S. Madachy asserts that
1)With a knowledge of the elementary properties of numbers
2) a simple desk calculator
The last 10 digits of this fantastic number (and other bigger nos.) have been calculated.
For the last 10 digits of 9^9^9 these have been calculated and are 2,627,177,289.
Can any one give me a method with the above conditions?
Note 9^9^9 is not equal to 9^81
mfgoode

[ET_]

Originally Posted by mfgoode

Mind Boggling Number.
The largest number that can be written using only 3 digits is 9^9^9.
Mathematician and editor Joseph S. Madachy asserts that
1)With a knowledge of the elementary properties of numbers
2) a simple desk calculator
The last 10 digits of this fantastic number (and other bigger nos.) have been calculated.
For the last 10 digits of 9^9^9 these have been calculated and are 2,627,177,289.
Can any one give me a method with the above conditions?
Note 9^9^9 is not equal to 9^81
mfgoode

Maybe with modular arithmetic?

Is he that Madachy that has been chief editor of the Journal of Recreational Mathematics?

Luigi

[mfgoode]

Originally Posted by ET_

Maybe with modular arithmetic?

Is he that Madachy that has been chief editor of the Journal of Recreational Mathematics?

Luigi

Yes, He was editor but Im not sure if he still is.
Modular arithmetic probably is the answer but how do you go about it?I dont
know and only have a vague idea.
Its interesting to read the other replies to me.They seem to say that it is possible and theoretically this can be done. But how about short cuts to make this physically feasible in ones lifetime! How did Madachy work this out ? before the advent of computers that we are fortunate to have today?.
Mally