schroder said:
I am now very seriously considering building my own TT but it will have all the bells and whistles required to do detailed experiments. I will have at least three tachometers, one on the TT, one on the rotating crossarm, and one on the wheel.
Brilliant. I make a strong prediction. If T1 is the tachymeter (giving the number of rotations per second of the table) of the TT, T2 is the tachymeter on the arm (giving the number of rotations per second) and T3 is the tachymeter on the wheel of the cart (giving the number of rotations per second), then, if the radius of the track on which the wheel is running on the table is R, and the radius of the wheel is r, and the wheel doesn't slip, then I predict:
T3 = R/r * (T1 - T2)
The above formula is valid for all motions if T1 and T2 are *signed* tachymeters, that is if they give you a positive number if their object turns CCW, and a negative number if their object turns CW. (or vice versa)
If the tachymeters just give absolute values and no sign indication of the sense of rotation, then we have to use the above formula if both TT and arm turn in the same direction (CW or CCW), and we have to use:
T3 = R/r * (T1 + T2) if the table is going CW and the arm CCW, or vice versa.
The sign of T3 depends on which side the tachymeter is mounted on it and if it is a signed tachymeter.
I make the above prediction without any comprehension of the zeros of Bessel functions in the Superheterodyne Class A Reboosted Overdriven Dolby Surround theory of rotating tables, but just based upon the defining property of non-slipping wheels, that the point of contact has the same velocity on both sides of the contact, and the earlier given formula which you disputed.
Let experiment decide
EDIT:
Just to be more explicit, in post 762, I wrote amongst other things:
Second application:
Now, if the wheel is not on a road, but on a treadmill that GOES TO THE RIGHT with a velocity v_tread (positive number: the velocity vector of a point on the treadmill is (v_tread,0) and this is a vector oriented to the positive X-axis, so to the right), then, if the wheel is making a NON SLIPPING CONTACT, we see that the point at the bottom of the wheel is having the same velocity as the tread (as it isn't slipping there and in contact), so we have equality of the two velocity vectors:
( + w R + vx, vy) = (v_tread,0)
from which:
w R + vx = v_tread and vy = 0
In other words: w = (v_tread - vx) / R.
and no vertical motion.
here w was the angular velocity of the wheel, vx was the velocity of translation of the cart, and v_tread was the velocity of the surface, both velocities in the same direction of course. R was the radius of the wheel, which is now to be called "r".
So, we take this expression to be w = (v_table - v_cart) / r
Now, if we take this direction to be "right to left" on the turntable(*), when the cart is nearby the observer, then we have:
v_table is (2 Pi R T1) (circumference of the track, times the number of times this table turns per second)
v_cart is (2 Pi R T2) (circumference of the track, times the number of times the arm (and hence the cart) turns per second)
w is 2 Pi T3 because 1 turn per second (unit of T3) comes down to 2 Pi radians per second.
So we fill in: 2 Pi T3 = ( (2 Pi R T1) - (2 Pi R T2)) / r
We can bring out 2 Pi R of the numerator of the fraction on the left side:
2 Pi T3 = 2 Pi R (T1 - T2) / r
We can divide by 2 Pi (2 Pi is not zero) on both sides of the equality:
T3 = R (T1 - T2) / r
We can change the order of the factors in a product (commutativity of x in R,+,x) :
T3 = R / r (T1 - T2)Tada Tadaaaa !
(*) Just to be completely clear: I just flipped the positive orientation of the X-axis, which was "left-to-right" in post 762, and which I now take "right-to-left" in this post, to be in agreement with the videos and all the conventions others have used up to now. The formula remains of course just as valid, except that w will now be positive in the CCW direction (we look upon the picture now from the other side).
