MAX EBB — L
ee Helm hardly seems to have time to crew for me on my boat these days. Between her windsurfer and her thesis she just doesn't seem to have any extra time. So for the last Midwinter race, I gave up finding my own crew and decided to accept a ride on a bigger and faster boat. And, as luck would have it, Lee showed up on the same boat for the same race. "Lee!" I confronted her. "I thought you said..." "Chill, Max. My midterm got moved, so it turns out, like, I was free this weekend after all." But the truth was that this boat is bigger and faster and newer than my boat, and probably serves better sandwiches too. After all, I had made essentially the same choice as Lee. She was given the trimmer's job, and I was assigned to forward 'pit. The race was going well, but with a new sea breeze filling in from a new direction, our first downwind leg had turned into a hot reach. Lee was trimming, so she moved the sheet over to a windward halyard winch from which she could see the curl of the spinnaker luff. Another crew jumped in to grind for her as the wind continued to build. "Another wrap, please," Lee asked the grinder. "Use the big winch!" shouted the skipper. That little halyard winch won't hold nearly as well as the primary. The big winch has a much bigger drum." The illustration below shows sheet tension and tailing tension, as well as the wrap angle on a winch's drum.
"Not true," she yelled back, without looking down from the sail. "Holding power is equal to E-to-the mu theta, where mu is the friction coefficient and theta is the angle of contact. The diameter, like, doesn't even...Grind! ...appear in the formula." "I think the big winch does hold better," I said, agreeing with the skipper. "More bearing area." "Do the math," Lee challenged. "Area is pi times diameter times height of the drum," I replied. "More area equals more holding power. Right? Right?"
I always have an uncomfortable feeling when Lee tells me I'm wrong about something. "Wrong," she stated flatly. "Derive the formula and see what you get. I always have an uncomfortable feeling when Lee tells me I'm wrong about something I had always held to be true. But there was no time for more debate, the leeward mark was coming up fast. Once we were cleaned up from the douse and mark rounding, I found myself sitting just aft of Lee on the rail for the long beat back to the top of the course. "Time for our math lesson," Lee warned as she took out a small notebook and a pen. "What, you're going to prove that winch diameter doesn't matter?" said one of the several foredeck crew who were sitting on the rail just forward of Lee on the opposite side of her from me. "It's an easy proof," she said. Here's the setup." She drew a top-down picture of a winch drum with a rope running most of the way around it. "Clockwise, stupid!" joked the foredeck crew when he saw that she was drawing the sheet load force on the right and the tailing force on the left. Lee quickly corrected the faux pas. The reversed diagram was much more to our liking.
"Here's the tension from the sheet load, here's the tension on the sheet tail, and theta is the angle of contact on the winch, in radians. I'm only showing three-quarters of a turn, but if it goes around a full turn that's two-pi, three turns would be six-pi, etc." "Don't you want to use R for the radius of the winch?" I asked. "Don't need it, and you'll see why. Let's look at a small but finite section of the sheet as a free-body diagram."
N
ext she drew figure two, showing only a small part of the sheet, in contact with the winch drum through only a small contact angle. "The angle of contact is delta-theta. The force on the right side is T, acting downward on the diagram at an angle of theta over two. The force on the left side — remembering that we have a clockwise winch, is T plus delta-T, also acting down at an angle of theta over two. There's a friction force F, acting tangentially away from the sheet load, and there's a normal force N, pushing up." "Shouldn't the rope be pushing down in your diagram?" I asked. "No, it's a free body diagram. We're only looking at forces acting on the sheet. The winch pushes up on the line. Let's add up forces in the X direction, and we get: (T + T) Cos( /2) = F + T Cos( /2) "The angle from horizontal is deltatheta over two, the horizontal component is the cosine of that angle. We have T plus delta-T on one side, and the frictional force plus T on the other side." "Fair enough. Now you do the vertical forces?" "For sure. We have:" (T + T) Sin( /2) + T Cos( /2) = N "Where N is the normal force acting up on the line. It has to be equal to the downward components of T and T plus delta-T, both times the sine of one-half theta." "I'm still with you," I said. "Now make the theta angle small, but
