Looking at the old cassette tape,
From the POV of the head, let's say that it reads at speed \$v\$ (the magnetic medium scrolls at speed \$v\$).
But looking at the right wheel, which is the one that's pooling the magnetic medium – its radius is growing(!) over time.
Now, \$v=r\omega\$, where \$\omega\$ is the angular velocity, i.e. a constant
Question
I don't think that's true. What is really going on here? Radius is growing over time, for sure. I also assume that \$\omega\$ is constant. so did \$v\$ increase?
Best Answer
The details of how a cassette drive works are well covered by this Wikipedia article. The tape is pulled by a capstan next to the playback head, and this capstan pulls the tape at a steady rate.
(picture from the Wikipedia article)
You probably need to click on the picture to see it full size. I have indicated the capstan by a red arrow. The take-up spool doesn't rotate at a fixed speed. It uses a slipping drive, as badjohn says in his answer, so it takes up the tape at the speed the capstan moves it.