Taperecorder mechanics

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In summary, the mechanics of a tape recorder involve the recording head passing with constant velocity, causing the spools to rotate with decreasing velocity as the empty wind is filled. To find the spools' angular velocity, the tape's width (b), radius of the empty wind (r_0), velocity at the recording head (v_0), and time (t) are needed. The angular acceleration is negative due to the decrease in angular velocity while the radius increases. To solve the problem, one must work out \omega = d\theta/dt in terms of r and differentiate to get acceleration. One possible solution is to write the equation as dr=\frac{b}{2\pi}d\theta and divide by dt to
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N2
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hello, I've just started a mechanics course, and i don't quiet know how to start with this assignment:

In a taperecorder the information to pass the recording head with constant velocity. Therefore is has to rotate with decreasing velosity as the empty wind is getting filled.

What is the spools angular velocity, [tex] \omega [/tex], as function of
[tex] b [/tex], the tape's width
[tex] r_0[/tex], radius of the empty wind
[tex] v_0[/tex], velocity at the recording head
[tex] t[/tex], time

My first thought was, that the angular acceleration must be constant, but how does this help? and how can i solve this problem?

thanks
 
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  • #2
For clarification, I believe that b is the tapes thickness. So in that case, the radius is increasing by b with each revolution, or dr = b.

The linear velocity must be constant, so the angular velocity [itex]\omega[/itex]= v/r must be decreasing as r increases (proportional to r). If the angular velocity has to decrease, what does this say about the angular acceleration?
 
  • #3
You are right; b is thickness.
If the angular velocity decreases then the acceleration is negative. But is your point that a. acceleration, [tex]d\omega[/tex], is proportional dr and therefore to b ?
 
  • #4
Astronuc said:
So in that case, the radius is increasing by b with each revolution, or dr = b.
dr is only an infinitisimal quantity, while b is finite. I think you mean to say [itex]\frac{dr}{d \theta} = \frac{b}{2 \pi}[/itex]. This seems to be a good way to start the problem.
 
  • #5
LeonhardEuler said:
dr is only an infinitisimal quantity, while b is finite. I think you mean to say [itex]\frac{dr}{d \theta} = \frac{b}{2 \pi}[/itex]. This seems to be a good way to start the problem.
Something like that, yes.

So one has to work out [itex]\omega[/itex] = d[itex]\theta[/itex]/dt in terms of r and differentiate to get acceleration.

Therefore is has to rotate with decreasing velosity as the empty wind is getting filled.
Implies that the angular acceleration is negative, because angular velocity must be decreasing while the radius is increase in order to maintain a constant linear velocity.
 
  • #6
I don't think finding the acceleration is important to this problem. It asks to find [itex]\omega[/itex] in terms of the other variables. The way I solved it was to write the equation as [itex]dr=\frac{b}{2\pi}d\theta[/itex] and then divide by dt to make [itex]\omega[/itex] appear. Then after separation of variables, integration, etc. you get an answer. Maybe there is a simpler way, though.
 
  • #7
LeonhardEuler said:
... divide by dt to make [itex]\omega[/itex] appear. Then after separation of variables, integration, etc. you get an answer.
integrating the equation containing [itex]\omega [/itex] - wouldn that just give you an expression for [itex]\theta[/itex] ?
 
  • #8
N2 said:
integrating the equation containing [itex]\omega [/itex] - wouldn that just give you an expression for [itex]\theta[/itex] ?
Not the way I did it. I re-wrote [itex]\omega[/itex] as [itex]\frac{v}{r}[/itex]. Then, since v is constant, I just separated variables and got an expression for r in terms of t.
 
  • #9
ok thank you.
the expression i end up with is now
[tex]\omega (t) = \frac{v_0}{ \sqrt{b/ \pi \cdot v_0 t + r_0}}[/itex]
but isent there i dimension problem with the units?

[tex][s^{-1}] = \frac{[m/s]}{\sqrt{[m] [m/s] + [m]}}[/tex]
dosent seem to add up? (the problem being [itex] r_0 [/itex])
 
  • #10
Yes, re-check the work you did to find the value of the constant after you integrated. I get [itex]r_0^2[/itex]
 
  • #11
Another way might be to start with both reels with the same amount of tape, both moving the same velocity, say 1 inch per second at the recording head.


Assume 1 ips, then what's the starting radius? It can't be the center because there is a hub. Imagine its exactly 1" radius from the center of the reel. So at 1 ips, with 1 inch from center, assume it can never change velocity because the reels are always equal, to get an initial calculation.

If you start with a 1" radius you would add the tape thickness and recalculate for every revolution. The other reel isn't necessarily going the inverse speed, if that reel starts its overlap at some different part of rotation.

The speed of the reel changes on the thickness of the tape. Constantly? Or only at the time the tape overlaps? It has to be when the tape overlaps, it accelerates up to the thickness of the tape then is constant until the next revolution.

Imagine the reel is 5 feet in diameter and the tape is 1 inch thick, it winds on to the takeup side then overlaps, now the radius is changed by 1 inch and moves slower. It would probably increase, overshoot, then correct each rotation to the current radius +1 tape thickness
 
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What is a taperecorder?

A taperecorder is an electronic device used for recording and playing back audio. It uses magnetic tape to store the audio information.

How does a taperecorder work?

A taperecorder works by using a recording head to convert sound waves into electrical signals, which are then stored onto the magnetic tape. During playback, the signals are read by a playback head and converted back into sound waves.

What are the main components of a taperecorder?

The main components of a taperecorder include the recording head, playback head, motor, capstan, pinch roller, and magnetic tape. The recording head converts sound waves into electrical signals, the playback head reads the signals from the tape, the motor drives the tape, the capstan maintains tape tension, and the pinch roller presses the tape against the capstan.

What are the different types of taperecorders?

There are three main types of taperecorders: reel-to-reel, cassette, and digital. Reel-to-reel taperecorders use large reels of tape and are generally used for professional recording. Cassette taperecorders are smaller and use compact cassettes for recording and playback. Digital taperecorders use digital technology to record and store audio.

What are some common issues with taperecorders?

Some common issues with taperecorders include tape damage, motor malfunction, dirty or misaligned heads, and mechanical failures. These issues can affect the quality of the recorded audio and may require repairs or replacement parts.

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