Usually, the needle move from the edge of the phonographic record towards the center. The distance travelled by the circle can therefore be calculated by summing the circumference of the circles traced by the needle.
Thus, one needs to determine the distance of needle from the center of the phonographic record at various times, corresponding to each circle traced by it.
(a)
To calculate the position of needle relative to the point fixed in a room, one has to first determine the position of center of rotating record, and the position of the needle from the reference point. Then using these distances, one can calculate the radius of the circle traced by the needle at all times, and eventually the distance travelled by the needle.
(b)
When the reference point lies on the rotating frame, the point itself moves with the rotating record, and would see the record at rest, but the needle moving in around the record.
Therefore to calculate the distance travelled by the needle, the observer would need the distance between the center of record and the needle at various times. From his knowledge, the observer can simplify the problem to rotation of needle in circular paths around the center of the record, and use the distance between center of the record and needle to calculate the total distance travelled by the needle.
(c)
When the reference point lies on the arm of the record player, the arm would appear to be stationary, and the record would appear to rotate. The observer would also notice the change in the distance between the center of the record and the position of the needle, and to calculate the desire, the circumference of the circle corresponding to various radiuses must be added.