first let me get some things out of the way. I'm a programmer at heart, but not very versatile with the maths stuff. Which is why I come here to seek help.
That said, I don't fully understand everything regarding formulas and integration etc, when it comes to this problem. So if possible you could also express the steps or solution in programming terms, I'd be very grateful!
The problem
I'm trying to find a solution to a derivative problem of a, what seems to be, a classical mechanical problem.
The original problem is finding the final angular velocity between two discs, spinning at different speeds, after they've touched eachother. Think a record player and a record being dropped on top. As such the problem focuses on conservation of angular momentum as far as I have researched.
Now, I've come as far as to solve for the final angular velocity of the two discs; that solution and formula is readily available with a search.
But I can't find the solution on how to integrate this over time.
For instance, how do I know how long it will take the discs to get to final velocity?
How can I integrate this with a defined time step?
What I've got so far…
Given that the discs have intertia $I_n$ and an initial angular velocity $\omega_n$, and the systems total inertia $I_t = I_1 + I_2$
then the final velocity for both discs should be
$$\omega_f = \frac{I_1\omega_1 + I_2\omega_2}{I_t}$$
So what I've figured out is that, obviously the total difference in angular velocity for a disc would be $\Delta\omega_n = \omega_f – \omega_n$ and that's where I start to get lost.
If I'm understanding it right, the total difference in angular velocity can be seen as an instant impulse to acceleration?
How do I integrate this acceleration over time?
What I've got so far is, if we have angular velocity $v$, and timestep $t$ then either,
$$\alpha_n = \frac{\omega_f – \omega_n}{I_n}$$
or divided by total inertia,
$$\alpha_n = \frac{\omega_f – \omega_n}{I_t}$$
and then I integrate like so
$$\omega_n = \omega_n + {\alpha_n}t$$
Am I right in my assumptions and/or calculations?
What did I get wrong if not? What's the proper way to integrate the acceleration and velocities of the discs, over time?
And finally
One last thing I've been pondering about when thinking about this problem is; how does friction and outside forces (i.e. torque) come into play in these formulas/calculations? How would I calculate it?
If you've come this far, thank you for taking your time to read through my problem. If you don't have a solution I'm still thankful for reading it.
If however you know the solution, please let me know! This problem has been bugging me for a while, and I can't seem to find any solution to it at all. Only for final velocity.
Thank you!
Best Answer
The situation seems basically similar to a "dry clutch" in engineering. The simplest model of that, I think, would be almost identical to a block sliding on a flat surface, subject to frictional forces. And the common assumption there is that the kinetic friction force is proportional to the normal force applied to the block (which might be its weight), and to a coefficient of friction, but it is independent of the speed. This is the angular equivalent of that. The frictional torque between the two discs will act on the relative motion, and will conserve total angular momentum.
So we can start by writing $$ I_1 \dot{\omega}_1(t) = -\tau , \qquad I_2 \dot{\omega}_2(t) = +\tau $$ where the dot represents the time derivative and $\tau$, the frictional torque, is a constant. We need to integrate these equations in time, from $t=0$, up to the point where $\omega_1(t)=\omega_2(t)=\omega_f$. Beyond that point, the discs will rotate together.
This is simple enough to do analytically. The angular velocities will change linearly in time $$ \omega_1(t) = \omega_1(0) -\frac{\tau t}{I_1 } , \qquad \omega_2(t) = \omega_2(0) +\frac{\tau t}{I_2 } $$ An easy way to get a solution is to write an equation for the relative angular velocity $\omega=\omega_1-\omega_2$, set $\omega(t)=0$, and solve it for $\tau t$: \begin{align*} \omega(t) &= \omega(0) - \tau t \left(\frac{I_1+I_2}{I_1I_2}\right) = 0 \\ \quad\Rightarrow\quad \tau t_f &= \left(\frac{I_1I_2}{I_1+I_2}\right) \omega(0) = \left(\frac{I_1I_2}{I_1+I_2}\right) [ \omega_1(0)-\omega_2(0)] \end{align*} where I've called this final time $t_f$. We can check that this gives the correct answers by substituting back in \begin{align*} \omega_1(t_f) &= \omega_1(0) -\frac{1}{I_1 }\left(\frac{I_1I_2}{I_1+I_2}\right) [ \omega_1(0)-\omega_2(0)] = \frac{I_1\omega_1(0) + I_2\omega_2(0)}{I_1+I_2} \\ \omega_2(t_f) &= \omega_2(0) +\frac{1}{I_2 } \left(\frac{I_1I_2}{I_1+I_2}\right) [ \omega_1(0)-\omega_2(0)] = \frac{I_1\omega_1(0) + I_2\omega_2(0)}{I_1+I_2} \end{align*} So, both equal to $\omega_f$ at that time $t=t_f$. Of course, if you want to make the frictional torque depend on relative angular velocity in some complicated way, the solution may require a computer. But the underlying equations will be similar to the above.
[Edit following OP comments]
Note that I wrote the equation for $t_f$, time needed to reach equal angular velocities, as an expression for the product $\tau\, t_f$: this product is equal to a function of the initial angular velocities and the moments of inertia. You need to know $\tau$ before you can calculate $t_f$.
To evaluate $t_f$ for a particular physical case, you need to multiply my equation for $\tau t_f$ by $1/\tau$ on both sides, i.e. take $\tau$ over to the right hand side. This illustrates that $t_f$ is inversely proportional to $\tau$, if the other parameters are kept constant. If the friction between the two discs is zero, it will take an infinite time to reach the same angular velocity, because the two discs will have no effect on each other. If the friction is very large, the time taken will be very short. In any case, the integration of those equations should stop at $t=t_f$, since after that time the two discs are not rotating relative to each other.
So, to solve the problem you want to solve, you must provide the physics of the interaction between the discs. Let me emphasise that my solution is only based on the simplest assumption about this. Your situation might be more complicated. However, some aspects of my solution will still apply, e.g. the torque between the discs must still conserve total angular momentum. For more discussion of friction in general, see for example https://physics.stackexchange.com/questions/2408/does-the-force-of-kinetic-friction-increase-with-the-relative-speed-of-the-objec and https://physics.stackexchange.com/questions/154443/why-is-the-equation-for-friction-so-simple .