Expert: Scott A Wilson - 12/17/2008

Question
Let L be the circle in the x-y plane with center the
origin and radius 38.
Let S be a moveable circle with radius 14 . S is rolled along the inside of L without slipping while L remains fixed.
A point P is marked on S before S is rolled and the path of P is
studied.
The initial position of P is (38,0).
The initial position of the center of S is (24,0) .
After S has moved counterclockwise about the origin
through an angle t the position of P is
x = 24cost +14cos(12/7t)

y = 24sint −14sin(12/7t)

How far does P move before it returns to its initial position?
Hint: You may use the formulas for cos( u+v) and sin( w /2).
S makes several complete revolutions about the origin before P
returns to (38,0).

Answer
Since the factors of 38 are 2 and 19, it can be seen that only the 2 goes into the 24.

Lets take a smaller case.  Let R be the big radius of the fixed circle and r be the radius of the small circle within.

If we take R as 4 and r as 2, it can be seen to take 2 times for the little circle to rotate and only 1 revolution around the big circle was made.

If we take R to be 10 and r to be 2, it can be seen to go around 5 times and still only make 1 revolution.

Basically, whatever we need to multiply R by to be divisible by r, that's the number of revolutions it makes.

If we take R as 20 and r as 8, note that it will be complete in 2 revolutions.  2R = 40, and when 40 is divided by 8, we find that the small circle rotated 5 times.

Now, lets consider the problem at hand.  We have R ( the big radius ) is 38 and r (the small radius) is 14.  At the start we said that 7 was left over in the little circle after removing the common divisor, so we need to mulitply the big radius by 7, giving 266.
When 266 is divided by 14, it tells us how manu revolutions the little circle had.

So the revolutions of the little circle is 266/14 (you can do the math) and the big circle is gone around 7 times.

What to do is to determine the distance travelled by that particular point P until P touches the big cirle again.  This should be multiplied by 19 (yes, that's 266/14) to get the total distance travelled.

What needs to be done is an equation for that point P as a function of x and a function of y.  Once these equations have been found, the equation to find the distance is

⌠N
⌡0 √((dx/dt)²+(dy/dt)²) dt.

I have never even thought of find the postion equations of a point P on a small circle revolving around the inside of a big circle, but it can be done.  It must ave to do with the small radius, the large radius, and how far the cricle has travelled.

For some distance s, the number of radians moved the small circle is s/r and the number of radians moved on the large circle is s/R.  This is because we divide by 2π to find what percentage of the revolution it has made and multiply by 2π to get how many radians it has moved, thereby cancelling the 2π.

If you need to know what x(t) and y(t) are, you can write back again and I might have them figured out.