The Exponential Measure of Time

**Abstract **

When watching a movie on a video tape, people usually believe that the counter acts on a linear function of time. However, the counter actually behaves as an exponential function of time. As the reel gains tape, its radius increases. The velocity of the tape across the head is constant. Therefore, as the circumference of the reel increases proportionally to the increase of the radius, the angular velocity of the reel must decrease. After watching the tape twice, we graph the number on the tapecounter verse's time. Through separable differential equations we are able to determine the equation for the relationship between the counter and time. Finally, we check the graph of the equation against the graph of our original data. Quantitative and qualitative analyses of these graphs reveal that our derived equations accruately portray the relationship between counter readings and time.

**Introduction **

This lab analyzes data from a VCR tape deck with digital counter (example 4.20 in our textbook). We attempt to model the relationship between tapecounter readings and time. Making our data runs, we start with the counter at zero and a fully rewound tape of NOVA's Einstein. As the tape is playing, we record the number on the tape counter every five minutes for the duration of the movie. We then use the data to determine properties of the tape and VCR.

Additionally, we take measurements
on the tape to determine actual *R*(0), and with this data
we also determine tape thickness, *c*, by measuring a given
wrapped radius, subtracting *R*(0), and dividing by the number
of turns on the tape. Once the length of the tape is determined,
*v* follows from the average of the length divided by time
as *v* is maintained constant by the machine.

We begin by analyzing the problem.
The tapecounter tracks the number of revolutions of the take-up
reel. By noting that the number of revolutions is proportional
to the angle, *A(t)*, we can relate tapecounter readings,
*n(t)*, to time by the equation

A relationship between the angle
swept out, *A(t) *measured in radians*,* and the radius,
*R(t), *can be seen in Figure 1.

Figure 1 The take-up reel (seen from above).

The area of the tape on the reel
(as seen from above) at time *t* is given by the washer-shaped
region in Figure 1, p[*R**2*(*t*)*
- R**2*(0)].
Dividing this area by the thickness of the tape, *c*, we
obtain the amount of tape on the reel at time *t*, which
is also given by the velocity times time, *vt*, where

The angle swept (measured in radians)
times the radius again gives the amount of tape on the reel at
time *t*, so that the velocity can be written as the change
in *A(t)* with respect to *t* times *R(t). *Solving
for angular velocity, *dA(t)/dt*, gives

(3)

Solving equation (2) for *R(t)*,
collecting constants, and substituting into equation (3) gives
the differential equation

(4)

where the constant *b* is given
by

(5)

The solution of equation (4) is found by separating variables and integrating both sides

(6)

Using the initial condition* A*(0)=0,

* C*
= ,

which gives an equation for the area with respect to time,

(7)

Substituting equation (7) into equation (1) and defining

* *

shows the tapecounter readings, *n*(*t*),
are a function of *t* and the constants *a* and *b*

(8)

Here the constants *a* and *b*
depend only on the tape and the machine being used. This equation
should model the data (refer DATA1). Comparing graphs of the
data to graphs of the model should substantiate the validity of
this model. **Results and Discussion**

** **Validity
of the model is indicated by the correlation of the data's best
fit curve to the model's curve.

In order to gleen more information
from equation (8) as it relates to the machine and tape, we solve
for *t *and divide through by *n* to get the linear
equation

(9)

Plotting equation (9) allows reading
off the slope and y-intercepts and solving for *b *and *a*
(Figure 3). Figure (3) shows a slope of 0.00004705 and y-intercept
of 0.092352. Using the values for slope and y-intercept obtained
from Figure 3, Maple is used to solve for *a* (981.5) and
*b* (.02206). Figure 4 is a plot of the data superimposed
on the curve from equation (8) with these values of *a* and
*b*.

**Conclusions**

Quantitative analyses of Figure 4 reveals close fit (2.5 percent error) between the model and data. Based on this we feel confident that we could use this model to predict VCR/tape properties, such as tape length verses counter reading, tape velocity, and spool thickness verses time.