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 length of tape on the take-up reel, is constant. Let R be the initial radius of the take-up 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 b) Find a formula for . Your formula may have any or all of the following in it: n, t, R, τ 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.