"I thought of that while riding a bicycle."
For the last four days, I've biked across over 99% of Virginia. After dipping my bike's rear tire in the Atlantic Ocean in Yorktown, I'm here writing this post in a motel in Breaks, a small village just a mile east of the Virginia-Kentucky border.
I have to say, though, the last two days were substantially easier than the first two.
During the first two days, I found myself getting tired very quickly. It was an unusually short amount of time before I started counting the miles, dreading each stroke of the pedal, and wondering why the road always seemed to feel slightly steeper than it looked.
Then, I figured it out.
The Siva Cycle
One of my friends on the trip installed the Siva Cycle, intended to solve our energy problems. The device, self-proclaimed as "a lightweight, highly-efficient bicycle generator", renewably charges a USB battery pack as the rider pedals.
The rhetoric in favor of installing it was clear: so long as the Siva was on the bike, we would never run out of battery. Biking for an hour produces a full charge on the iPhone 6, my current phone, and we would be biking ten times that amount every day.
More importantly, the amount of drag it causes seemed reasonable. I asked my friend, who did his due diligence before buying the device, about how much resistance it adds. He told me not much, and indeed the Siva FAQ reaffirms this:
"The resistance is very small, ... [i]n our use around town, it is very difficult to tell that it’s on the bike.
I was so hesitant to think that my exhaustion was due to The Siva, that I tried finagling around with everything else first.
Was it the extra weight I was carrying? Or perhaps it was all in my head, from being nervous at my first cross country trip?
Nope. With all other explanations exhausted, I finally disassembled the Siva in the middle of the road. Then, like clockwork, my legs felt stronger, the bike felt faster, and biking across the country seemed like a reachable dream once more.
On a light day, we decided to test empirically exactly how much resistance the Siva produced. We ran 5 trials of pushing the pedal down with constant acceleration a half pedal cycle for intervals of 0.5s, 1s, and 2s. We then measured how many cycles the rear wheel turned before the bicycle stops.
Here are the results:
Half cycle at 0.5s
Average without Siva: 57.1 rotations
Average with Siva: 3.9 rotations
Half cycle at 1.0s
Average without Siva: 21.2 rotations
Average with Siva: 2.1 rotations
Half cycle at 2.0s
Average without Siva: 7.2 rotations
Average with Siva: 0.9 rotations
Yeah, its clear that you can't call this "very small" resistance.
There's a complicated, very technical way of solving for the resistance on the Siva Cycle that I went through with vector calculus. I'm not going to go through that here. Instead, to my delight, there's an elegant way to calculate the drag with only high school level physics.
We can model the cycling of the wheel as a free falling object initially launched into the air. Instead of:
We have the torsional equivalent:
The parallel is a such:
With freefall, v0 represents the launch speed
With cycling, θ0 represents the initial speed of the wheel after the manual spin
With freefall, a represents the acceleration of gravity that slows the object down
With cycling, Fi/Iwheel is equivalent to the initial applied torque, divided by the inertia of the wheel, equaling the angular drag.
In both freefall and cycling, time represents the amount of time for the object to stop.
Turns out, despite the farfetchedness of the comparison, it works.
Rearranging for F, we can calculate the drag force to be:
Since inertia is defined as such:
and never changes across trials, the drag force is inversely proportional to the time it takes for the wheel to slow to a complete halt. Initial velocity, because of the controlled nature of the spins, is also not changed. We can also assume time and number of cycles is directly proportional.
Meaning that at 0.5s, the Siva increases drag 214-fold; at 1s, its 104-fold; and at 2s, it is 64-fold.
Dear Siva, I'll be biking the rest of the trip without you.