If given the exact diameter of the spindle, speed and thickness of the tape, and gear ratio between the takeup reel and counter, it would be straightforward to come up with a way to relate the counter to time the tape has been playing and/or recording. The problem is how to derive this factor without it. All of these come out in the final conversion, in fact, it is not necessary to even use pi in this problem except to factor it out in the first step.

Let's assume a constant tape speed and that the tapes will always be rewound (and counter zeroed) before using. If we know the counter value after recording for exactly 1 hour and after exactly 2 hours, would that be enough information? Certainly. There is a direct ratio between the cross sectional area of tape on the takeup side and time as well as between the counter and change in radius of any "donut".

Since the counter starts with a zero value, the ultimate equation will be

1) Time = [(x + R)^{2}
- R^{2}] / K

where the **K** is a conversion factor, **x** is a counter value, and **R**
is the radius of the spindle in counter units. In the diagram, the red
area represents the tape for the 1^{st} hour while the yellow area does
the same for the 2^{nd }hour. Both have the same area--the yellow
donut is bigger but also thinner. We can equate these donuts and solve for
**R**, or assume two red donuts equal one red AND yellow donut.
Everything in the equation would be divided by the conversion factor **K** so
it isn't even shown (yet!).

2 * ( [x_{1} + R]^{2} - R^{2})
= (x_{2} + R)^{2} - R^{2}

2 * ([x_{1}^{2} + 2x_{1}R + R^{2}] - R^{2})
= x_{2}^{2} + 2x_{2}R + R^{2} - R^{2}

2 * (x_{1}^{2 }+ 2x_{1}R) = x_{2}^{2} +
2x_{2}R

2x_{1}^{2} + 4x_{1}R = x_{2}^{2} + 2x_{2}R

4x_{1}R - 2x_{2}R = x_{2}^{2} - 2x_{1}^{2}

R(4x_{1} - 2x_{2}) = x_{2}^{2} - 2x_{1}^{2}

R = (x_{2}^{2} - 2x_{1}^{2}) / (4x_{1} -
2x_{2})

Since we have **R**, we can plug it into Time = [(x + R)^{2} - R^{2}]
/ K as well as x_{1} which is the counter reading for one hour. We
already know Time = 1 hour so we can rearrange: **K** = [(x + R)^{2}
- R^{2}] / 1 hour. The number we get is our conversion factor and
includes everything mentioned in the 2^{nd} paragraph above.
Dividing [(x + R)^{2} - R^{2}] by our K will give the hours for
any counter value--further dividing K by 60 will yield the time in minutes in
equation 1). To determine the counter for a given time, subtract the
minutes from both sides of the equation and solve the quadratic for a positive
x.

Note
this diagram is an isomorph of the first one. The areas are defined by
subtracting the area of one triangle (instead of circle) from another and it is
still necessary to subtract R^{2} from (R + x)^{2}. Once
½ * slope * conversion factor is divided out, it is the same problem.

To run the webpage, please click VCR.htm

In another forum, Fetika created this diagram for R_{1} = 500 and R_{2} = 950 (or x_{1} = 500 and x_{2} = 950).