Flying First Class

In a little more than a week, I'll be joined in France by Marika and Lorna! This weekend I decided to pick up some supplies, like dog food and more bedding, and I was all over town yesterday. Most places are closed on Sundays here, so I was glad to have a nice rolling cart to carry everything at once:

Notice the three wheels arranged in a triangle – This allows the bag to to climb stairs, a helpful feature when you live 3 flights up! The three wheels rotate when the front one hits the face of a step:

I was curious how much this reduced the force that needed to be applied to climb the stairs. We can consider one step as the center joint rotating around the front wheel:

F is the force applied to the handle at an angle φ from the horizontal, and mg is the force of gravity pointing down. These each apply a torque to the joint:

where r is the length of the spokes. To climb a step, the wheels rotate 120° around the joint so the upper wheel lands on the next step. The question is, what are the values of θ that the wheel rotates through? We can simplify things if we make φ match the slope of the stairs. Then, after a bit of geometry that gave me middle-school flashbacks, we have that θ is in the range

If we apply just enough force to overcome gravity (𝛕 = 0), the force is

We can plot this for a couple different slopes:

The vertical lines represent the starting θ, and then you follow the curve to zero. The slopes are 15° in red, 37.5° in blue, and 52.5° in green. I think there must be a mistake in this model, since more vertical slopes require less force, even though a straight lift requires a force of 1 on this scale, the dotted line. Despite the effort saved by this invention, I'm still too worn out to find my error!