Finding angle as a function of time

[naphtalene]

Homework Statement

I don't know exactly how to integrate in this case.
This problem is about the track (spiral) of a spinning CD in a player.
The radius of the spiral depends on time/revelations and is given by:

$r=r_i+\frac{h\theta}{2\pi}$ , which means that the radius increases with h per revelation.

v ist the constant speed with which the disc surface passes the laser. The rate of change of the angle is given by:
$\omega = \frac{d\theta}{dt} = \frac{v}{(r_i + \frac{h\theta}{2\pi} )}$

I am looking for the angle as a function of time.

$\int \! \omega \, dt = \int \! \frac{v}{r_i+\frac{h\theta}{2\pi}} \, dt = ?$

Homework Equations

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The Attempt at a Solution

Well, I tried to substitute $\theta$ with $\omega t$ but this is definitely wrong because neither the angular velocity nor the angular acceleration is constant.
Now I'm really stuck ...I'm somewhat slow on the uptake tomorrow. Any hints? How can I solve this?

 

[gneill]

Re-arrange your equation to isolate dt on one side. Note that v is a constant so treat it as such. You should be able to integrate both sides of the resulting equation, one with respect to dt and the other with respect to dθ, with suitable integration limits.

 

[ehild]

$r=r_i+\frac{h\theta}{2\pi}$ (1), which means that the radius increases with h per revelation.

v ist the constant speed with which the disc surface passes the laser. The rate of change of the angle is given by:
$\omega = \frac{d\theta}{dt} = \frac{v}{(r_i + \frac{h\theta}{2\pi} )}$

I am looking for the angle as a function of time.

Differentiate equation (1) and substitute v/r for dθ/dt.You get a separable equation for r(t). Solve and use in (1) to get θ(t).

ehild

@gneill: The same thread at the same time again...

 

[naphtalene]

Well, if I rearrange this equation and differentiate it ...I get zero? I still don't get it =/

 

[gneill]

Well, if I rearrange this equation and differentiate it ...I get zero? I still don't get it =/

Can you show the rearranged equation?

 

[naphtalene]

Okay, I think I'm now a little step further.

$k = \frac{h}{2\pi}$

$\frac{d\theta}{dt}=\frac{v}{r_i+k\theta}$

$dt=\frac{r_i+k\theta}{v}d\theta$

$t(\theta)=\int_0^t \, dt= \int_0^\theta \! (\frac{k\theta}{v}+\frac{r_i}{v} ) \, d\theta = \frac{k\theta^2}{2v}+\frac{r_i\theta}{v}$

But what now?

edit: solving the quadratic equation gives me

$-\frac{2\pi(r_i+\sqrt{r_i^2-hvt\pi^{-1}})}{h}$

 

[gneill]

If you've solved for t as a function of θ, then you also have θ as a function of t by suitable rearrangement.

 

[naphtalene]

$\theta(t) = -\frac{2\pi(r_i+\sqrt{r_i^2-hvt\pi^{-1}})}{h}$

Can this be right? Seems kinda odd to me, but this is the only way I see to solve for θ

 

[gneill]

It looks reasonable to me.

[EDIT]: There are two roots to the quadratic, so you should verify that you've chosen the one that makes physical sense. Perhaps plug some representative values into the variables.

 

[naphtalene]

Great :) Then I think I finally got the clue... just need a little more experience :) Thanks!

 

[naphtalene]

Oh, I got another question.

To find the angular acceleration, I need to differentiate the angular velocity. But how do I do this here?

 

[gneill]

Oh, I got another question.

To find the angular acceleration, I need to differentiate the angular velocity. But how do I do this here?

You've just derived an expression for θ(t), the angular position with respect to time. What's the relationship between position, velocity, and acceleration?

By the way, I added a note to post #9; you should verify that you've selected the right root of the quadratic when you solved for θ.

 

[naphtalene]

Okay, I got it now. I differentiated θ(t) two times :) Thanks for your help!