Maths Problem-Cassette Player

Poorly defined question.

Like this:

When you play a cassette the tape unwinds from one roller unto the other roller.If you start playing a cassette from the beginning , can you plot the distance between the tapes [did you mean rollers here?] over the two rollers [then what is this?] over time?

Maybe OP should submit a graphic/diagram?

At constant rotational speed, with the wound material having consistent thickness, the OD of the wound roll is not linear with respect to time.

If you're asking if it can done, the answer is: yes it can be done If you're asking me if I can do it the answer is: yes I can. If you're asking how to do it the answer is: define more clearly what it is you want to know. A question had to be well - defined to get a useful answer. Your question does not clearly state what you want to know.

Seems clear to me. He asked if the amount of tape (distance) can be plotted over delta t.

Since it is a cassette winding at a standard 1-7/8 ips all you need is the bobbin size and the average thickness of the tape and you have everything you need (except the math formulas, of course).

Well, try this

A tape(sand) exits one side(the top) at a constant rate...

This is an interesting problem if I am interpreting it correctly. You want to calculate LEN as a function of time, given that R1 and R2 are both a function of time and given the linear velocity of the tape is constant. You will also need to specify the thickness of the tape in order to calculate R1(t) and R2(t)

This is similar to some of the calculations that I have done for web converting machines (unwind and rewind with constant web velocity)

This will give me something to play with today

The diagram is not a correct representation of a standard cassette. There are two posts that change the direction of the tape when running so that it is parallel to the centerline between reel axes. The angle of the tape with respect to an x-y axis at each reel axis with the x-direction coincident with the reel centerline will change as the length of tape wound onto each reel changes. In other words, the tangent point of the tape and the tape bundle on each reel will change with respect to the reel x-y axes. SO we need to calculate where the tangent point is so that the length of angles tape can be calculated for each reel. The center distance of the center reels is constant as is the distance between the direction-changing posts, so the length of the angles tape for each reel needs to be calculated to find the length of tape between reels.

That assumes the OP is asking for the length of tape between tangent points, and not the distance between tape bundles on the reel at the centerline. That's why I said the OP was not clear on what is being asked.

the speed of the motor might be constant but as one roll spools up its diameter gets larger and the speed of rewinding changes

Yes, but remember that the tape spools by at a constant rate (1-7/8 inches per second).

However, as one reel takes on tape the radial distance or size will not be linear. The circumference of the spool of tape gets larger with each revolution, based on the thickness of the tape.

Think of a roll of ribbon. So, the radial distance versus the amount of tape spooled is not a linear function, but it is easily calculated.

You need to know the diameter or radius of the take up reel without tape, the thickness of the tape, the total length of the tape, and a few simple geometry equations and you can calculate the amount of tape spooled for every revolution of the take up reel.

If memory serves, I think TDK tapes had a scale imprinted on the cassette body to give you an approximate time.

Unless it's one of these:

Go figure.

God I used to hate rewinding and re-taping the loop ends on those!!

Another interpretation of the problem. Same assumptions. This one is easier.

My Gut feeling is that the distance [DIST] will be constant over time, but I'll have to prove this conjecture

Clearly DIST(t)= D-R1(t)-R2(t)

That's also how I understood the question.

And the answer.

While R1 and R2 will change with time, the sum R1+R2 must remain constant, as tape can neither be created or destroyed. Therefore DIST= D-(R1+R2) must be constant.

Of course this is for an "ideal" cassette. For real life cassettes, the statement that tape cannot be destroyed is not necessarily true.

No, not constant. For example, if the supply reel is twice as large as the take up reel, when one layer of tape is removed from the supply reel, two layers of tape will be added to the take up reel, so the two will become 1 tape thickness closer together.

But, for only half the time. Then the process reverses.

So, it's a constant average, over time.

Don't forget that the spools increase or decrease in multiples of the tape thickness.

Yes! After I took the time to work this out ( instead of making some ill founded assumptions) I came up with the same results.

Good answer, the thickness of the tape, speed, diameter did not matter. I was just looking for the wave form of the distance between the tapes.

I will try to explain the answer without math. The three significant points are start middle and end. The dist. between the tapes are the same at the start and end.When it starts to play the distance decreases.The rate the distance changes is proportional to the difference of the rolls.When the rolls are equal, the distance will be the least and the rate of change will be at its least.

Also without maths..there are not two tapes under discussion.

There is only one tape, fixed at both ends........it's like timing a sprinter.. you have the total distance, and the timetaken to cross the winning line, so what does it matter what speed was being done around the bend?

I stand corrected R1^2 +R2^2 is constant but that does not imply that R1+R2 is constant.

This comes from the formula" L= (R^2 - R0^2)*Pi/T where: L= length of tape in a coil, R= radius of coil, R0 = radius of core, and T= thickness of tape

1

physics: cassette tape unwinding problem of spool 1 being unwound ..

I'm not even going to try to edit this:

given: spools 1 (left) & 2 (right) are free to rotate about their centers A & B; constant spool radii: r1 & r2; cassette tape of constant thickness e and constant total length S is wound around spool 1; end of tape from spool 1 is pulled & attached to spool 2; tape leaves spool1 tangent at pt. of departure C above A, making angle gamma from the vertical; tape meets spool2 tangent at pt. of entry D above B, making angle delta from vertical; tape around spool 1 makes total radius R from A to C; distance d (bet. A&B) > R+r2; spool2 unwinds spool1 w/ constant clockwise angular velocity omega2; define angle phi as angle between CD and horizontal w/ vertex at D; Find: angular velocity vector omega1(t) at any time t, angular acceleration vector alpha1(t), velocity vector v(t) and a(t) at C, phi(t), velocity vector d[phi(t)]/dt, acceleration vector d<sup>2[phi(t)]/dt<sup>2, R(t). pls. show solution, assumptions, conventions, differential equations or matrix solutions

There are several undefined parameters/question needed to be known to solve this nebulous question:

1. What distance between the one and only one tape surface is being measured?
2. What is the thickness of the tape?
3. What is the diameter of the empty spools?
4. What is the distance between the two spools at the axis of each rotation?
5. What is the total length of tape?
6. How precisely must this distance be known?

With these unknowns identified by either a numeric value or an algebraic character designation on the homework then an answer can be attempted.

It's obvious to me that the ratio of a full to empty spool diameter will determine the magnitude of multiple of tape thicknesses this will vary. I leave it to the student to derive the digital equations.

eltech,Now see what you have done!You've started another argument, and on a Sunday Morning yet! Oh, my.

I really enjoy math problems like this appearing on this forum. As I age, I have found my ability to analyse these problems mathematically is declining due to lack of practice. Without regular exercise, I fear that my ability will be reduced to the point where I have to take off my shoes in order to count past 10

I have to take off my shoes in order to count past 10.......Lovely!

My tape deck had a display on the front (in minutes and seconds) which stated exactly the tape's progress from spindle to spindle for the entire run. What's all the calculation about? Is this a trick question to detect senile decay?

CR4 provides that function quite well as evidenced by many of the postings, and sometimes even contributes to it

Cassette tapes come in 1.5, 1.0 and 0.5 mil thicknesses just like reel tapes.

Bullshit. There is no such thing as a recording tape of 1.5 mm thickness.

He said "mil" not "mm". A mil is 0.001 inch, a mm is approximately 0.040 inch. Like punctuation, units are important.

According to the great keeper of nearly useless information, Wikipedia, it seems that 90 minute cassette tapes are nominally 11.2 μm thick, about 0.5 mil. A 60 minute tape uses 16 μm thick media.

It is possible if you have a method of reading the relative speed of both posts simultaneously. Usually a sensor and a sensor plate that is attached to the moving post X 2 of some sort, fine steps...

Thats how may older VCRs calculate the position when in fast rewind/fwd, so as not to break the tape, it slows down as it gets near the start/end.....though I am certainly not an expert.

Most modern tapes areusually described as being so many millionths of a meter thick:-

Micrometer (British spelling micrometre), symbol µm. Sometimes called a micron.

It is known as 1 micron or micrometer with a mu symbol, then the metre symbol (µm)

Dear Mr.eltech,

About 35 years back, audio casettes have arrived, replacing records which will run at Speeds of 16 RPM, 45 RPM 78 RPM, The Cassettes Standards were of different sizes C45, C60 C90 and D120 45, 60 90 120 being the duration of the playing time in Minutes. Some cassete players will have a warning/note "Pl. do not play D120 in this player as it will cause damage."

Now I want to mention some thing about the speed of the tape. The speed of the tape at the Audio Sensing Head (termed as Capstan Head) as 1.125 Inch/Sec. which works out to 1.7149 Metres/Minute. When the cassette play is switched on, one side the spool is empty which will run faster in terms of RPM and the other one is full tape wound on it, which will run at slower in terms of RPM.

As the time proceeds, in spite of the CONSTANT SPEED of the TAPE the spool speeds get varied even though the tape is not driven by the capstan.

How the speed is varied by the driving spool. - is not known to me. I request the CR4 MEMBERS to explain the mechanism how the speed in RPM is adjusted.

DHAYANANDHAN.S

Slip clutch.

The original cassette design was by Philips and was adopted mostly world wide fairly quickly. The initial tape speed was 1.875"/second accomplished by a constant speed capstan shaft against a rubber roller, the take up reel (not a spool) is friction driven by the same motor that drives the capstan, the friction drive allows the take up reel to run at whatever speed necessary to keep slight tension on the tape as it is spooled onto the reel. The same thing happens in reverse play, rewind or fast forward except that the capstan roller is retracted.

There was a second cassette speed, 3.75"/second (double the 1 7/8"/second) which was used in some cassette players for better fidelity but reduced playing time by half. This speed was usually found only in the higher end decks.

The D120 cassette was not popular because many 'cheaper' cassette players could not handle the 0.25 mil tape, they frequently 'ate' the tape.

The cassette was used mainly as a portable convenience since reel to reel tapes tended to be larger, even though there were 3 1/2" reel tapes available which were equivalent to the cassettes in playing time. Technically they were audibly superior to the cassettes as the reel tape was 1/4" wide vs. the cassette's 1/8" width which forced the audio tracks to be much closer together which further increased 'bleeding' from adjacent tracks. It was originally not intended to be used as a high fidelity recording device, it was intended to be used for dictation and voice which required much lower fidelity.

About 35 years back, audio casettes have arrived, replacing records which will run at Speeds of 16 RPM, 45 RPM 78 RPM...

Interesting. In the US, the slow speed (long play) was 33 1/3 RPM. Did you really have 16 RPM in India?

16 2/3 RPM was a rare record in the US but they did exist. They were mostly audio books for the blind but they did exist.

Well I'll be..............!! Live and learn, as they say..

The distance once determined will not deviate over time.

As one roller diameter increases with added layers of tape the other roller decreases the exact same distance due to decreasing layers of tape on the roller diameter.

The speed of the driven roller is controlled so that the speed of the tape across the recording/reading head remains constant while the other roller is allowed to freewheel with a set amount of friction applied to avoid over-spooling and backlash.

The speed of the driven roller rotation determines how quickly the tape is depleted or increased on the appropriate roller in the selected/defined direction of tape travel.

The speed of the tape across the head is critical and specific to the model of cassette and any deviation from synchronous speed results in distortion of the recorded sound wave.

You must identify the diameter of the driven roller, determine the speed of rotation, identify the tape thickness, allow for the increase in roller diameter for each layer of tape, then do the math and plot the DATA points.

Yes, I can. It's a constant 7/8 inch per second. What's the prize?

There are a number of things you can look at with the cassette tape example:

Rotation speed of Supply Spindle over time

Rotation speed of Takeup Spindle over time

Speed of tape over Read-write heads (constant, set by Drive Roller)

Diameter of tape spool st Supply Spindle over time

Diameter of tape at Takeup Spindle over time

All the values that vary over time are simple 1st order Derivative Calculus problems. In fact, they are all the same problem: the example given in most calc textbooks is a water well with a bucket and rope on a cranked shaft, The rope forms a perfect spiral around the shaft, so the diameter at any given point is dependant on how many revolutions/partial revolutions the shaft has turned, as well as the speed the rope is playing out/winding up if the shaft is turning at a constand speed.

Once you look at the system clearly, it becomes a simple homework problem; And I'm sorry, but i do not do other people's homework problems for them.