Relationship between Counter and Time for a
Cassette Tape Player
Leslie Hogben
10/01
Most cassette tape players (and some old fashioned VCRs) have a counter that indicates
how many rotations of the take-up reel have occurred since the counter was last reset. You will
investigate the relationship between counter reading and time. We assume the counter is reset to
000 when the tape is fully rewound.
1. The table below gives data from a cassette tape player. Graph this data, with time as a
function of counter. Experiment with linear, quadratic and cubic regression to fit a curve to
the data. Choose one equation to describe the data. Justify your choice. Graph your equation
and data together. (The choice should be the lowest degree polynomial that gives a good fit.
For example, if a phenomenon is linear but includes some experimental error, the quadratic
and cubic regression equations will be essentially the same as the linear, and all 3 will fit the
data well. In this case, the linear should be selected.)
time
counter
0
000
5
037
10
069
15
097
20
123
25
146
30
168
35
189
40
208
45
227
2. Investigate why you obtained this type of equation: For the cassette player to function
correctly, the speed of the tape across the head must be constant. That means that if L is the
dL
is constant. Let R be the initial radius of the take-up
length of tape on the take-up reel,
dt
reel (it increases as tape winds onto it) and let τ be the thickness of the tape (R and τ are
constants). Let n be the counter reading. Observation has shown n = number of rotations the
take-up reel /6. Industry specifications require that the tape move across the tape head at 1.7
inches/minute.
a) What is the radius r of the take up reel as a function of n?
dn
. Your formula may have any or all of the following in it: n, t, R, τ
b) Find a formula for
dt
c) Find a function of t that has the formula found in (b) as its derivative (remember, R and τ
are constants).
d) Male sure your answers to questions 1 and 2(c) are consistent. Use your answers to
questions 1 and 2(c) to evaluate R, and τ.