Optimization question--of sorts

Optimization question--of sorts

I hope this isn't a silly question. I'm learning single variable calc, and
having lots of fun with optimization problems. This isn't exactly an
optimization problem, but something that came up while working on one.
Let's say I have a small circular garden with a short brick border. This
border is perhaps 1 foot tall, so that any sun or rain that reaches the
flowers has to come from directly overhead. Suppose that the radius from
the center of the circular flowerbed to the outermost edge of the circular
brick border is $r.$ I plant a metal rod at the circle's center. At the
top of the rod is a fan blade of sorts: it's flat, thin, parallel to the
ground, and has the shape of a circular sector with radius $r.$ This blade
is opaque, so it provides some shade for that part of the flowerbed
beneath it.
I give the blade a spin: as it's spinning, all of the flowerbed receives
some shade. Then I get an idea: I automate the spinning of the blade. I
can control the angular velocity, $w,$ of the blade with a remote control.
Let $l$ be the amount of light (or, if you want, rain) admitted to the
flowerbed. My question is this: Is it the case that $$\lim_{w\to
\infty}l=0?$$
I have reasons for thinking this is the case, and other reasons for
thinking it's nonsense. And if it is the case, then it's true regardless
of the value $\theta$ of the central angle of the circular sector, right?