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# Calculating surface area

Is there a formula for calculating the surface area of a ski's base? I would like to be able to compare ski sizes given their dimensions. For example, how would these two skis compare in terms of length feel relative to each other (assuming identical construction). a) 108-62-92,188cm b) 105-73-97, 186cm. Which would feel longer?
There are various formulae for the area depending on what degree of accuracy you require, and how many width measurements you want to make in addition to the usual tip-mid-tail data.

First, estimate the contact length, LC. If you have the skis available, you can just put the two skis bottom to bottom, and measure the distance between the contact points. If you don't have the skis available to measure, you can estimate the contact length as the published (chord length) minus about 15 cm.

Next, you need some width data. Lets assume you don't have access to the skis and just have the usual three published widths.

To a very good approximation, you can approximate the load bearing area of the ski as the sum of the areas of three rectangles:

1) a tip region which is about LC/4 long by W_tip wide;

2) a middle region which is about LC/2 long by W_mid wide; and

3) a tail region which is about LC/4 long by W_tail wide.

Thus, the entire formula is:

A_total = LC * (W_tip/4 + W_mid/2 + W_tail/4)

This formula is of course approximate, but it should give you absolute area numbers good to a few percent, and, perhaps even more importantly, allow you to get relative areas of one ski vs another to a very high degree of precision.

Hope this helps,

PM

PS - I just noticed your last sentence: "...Which would feel longer.."

The load bearing area of a ski really is important only in powder. The load bearing area determines the pressure the ski exerts on the snow, and hence how much you will compress (ie, sink into) soft snow.

Other quantities such as polar moment of inertia, sidecut, flex and flex distribution, initial camber, etc. are much more important in determining how "turny" a given ski feels in all conditions (incl powder).

There are, of course, ways to estimate all of these quantities, but they all require measurements that most private individuals would not be able to make, and/or detailed data which unfortunately is not published.

<FONT size="1">

[This message has been edited by PhysicsMan (edited June 22, 2001).]</FONT>
PhysicsMan writes:
> A_total = LC * (W_tip/4 + W_mid/2 + W_tail/4)

Gee. What a letdown. I was expecting somebody called PhyicsMan to define it as a parabola or some more complex curve and take the integral. We don't need no stinkin' linear interpolation. Issac Newton would be rolling in his grave!
ROFL!

I was just going to post that you should split the ski down it's longitudinal axis, apply a best-fit parabola and then integrate but I got beaten to the punch!

Geeks rock!
get the fatter ski
trust me
GeoffD & KevinH:

LOL guys!!!

Hey - I subscribe to the KISS principle, and I'm sure you all know what that is <G>!

FWIW, a long time ago, I once calculated what the differences between the simple rectangular area approximation (given above), and formulae like the trapezoidal rule, fitting a circle, fitting a parabola, etc, and they all came out to within a few percent of each other.

Perhaps more important, if you use any one of the formulas, the fractional error in the absolute area is almost identical for all skis (eg, the estimate for all skis are around 2% low), so the use of any one formula, even if very simple, doesn't change the rank ordering of the skis by area, or even the ratio of areas between any two skis.

If I get around to it, I'll look up my old notes and post the other formulae.

PM
PhysicsMan sez:
> Perhaps more important, if you use any one of the formulas, the fractional
> error in the absolute area is almost identical for all skis (eg, the
> estimate for all skis are around 2% low), so the use of any one formula,
> even if very simple, doesn't change the rank ordering of the skis by
> area, or even the ratio of areas between any two skis.

What??? My life is wasted! Three semesters of calculus, differential equations, 3 semesters of physics, probability theory, numerical analysis, 'bout 20 Double-E courses... all down the drain. I'm gonna go scrape the summer coat of wax off and sharpen up my X-Screams and slash my wrists with 'em.

I just looked up my notes.

For the assumptions I made in my rectangular approximation, the formula given before turns out to be identical to the Trapezoidal Rule for definite integrals (ie, straight line segments between the points). The reason is just because of the assumed way the ski got divided up - 1/4, 1/2, and 1/4.

Simpson's Rule assumes a parabola fit to the three points (as suggested). The formula for that approximation is:

A_total = LC * (W_tip/6 + 2*W_mid/3 + W_tail/6)

For a big 200 cm powder ski (185 cm assumed contact length), and with a 120-93-110 side cut, the two formulae give areas of 1924 and 1853 cm^2, respectively. This is a difference of 3.6%.

For a more shapely ski, the error will be larger. For example, lets take a 170 cm (155 cm assumed contact length) with a 105-63-95 sidecut. The two formulae give: 1263 and 1166 cm^2 for a 7.6% difference.

Geeks rule!

PM
Hey Geoff - Don't do that with your X-Screams - the edges will start to rust!

Maybe you can derive some consolation out of the fact that not one person in a hundred will have any idea what we are talking about - WE ARE THE ELITE - well, maybe not - we're just the crazed.

PM
Quoth PhysicsMan:
> Simpson's Rule assumes a parabola fit to the three points
> (as suggested).

Wrongo Buckaroo. Simpson's rule says it's time to go get a beer. (Homer Simpson, that is). It's Friday. I'm outta here!
DITTO!!!
TJazz- there are a lot of variables other than caculating the base surface area of a ski.All those formulas on the other posts, while intriguing, frankly will only complicate your life, and who needs any more of that.

Both skis are the same length, with the biggest difference being the waist width. That wider ski is not going to perform as fast edge to edge on the groomed, but will you give you better flotation in the crud, gunk, and POW.

So where do you ski the most, what type of conditions do you like to ski in/or on, and what do you aspire to in your future skiing accomplishements ? Answering that, and this will help you decide which is best to demo.

Surface area calculations or OK for the ski makers, and very sophisticated racers, racing teams, and totally obsessive and compulisive personalities. I think one of the other posts mentioned something about "keeping it simple."

So here it is: "Try'em before you buy'em!"

You are your own best judge.<FONT size="1">

[This message has been edited by wink (edited June 23, 2001).]</FONT>
Wink - I couldn't have said it better myself!

Because its my job, I either remember, or can look up in a minute, almost any physics / engineering formula that I need. Nevertheless, even I don't use formulas to help me decide between skis.

To people who aren't ski designers, but have some engineering / physical sciences background, I think the best use of such formulas is simply that they can be a general guide to the behavior of skis.

For example, consider the formula for the polar moment of inertia of a long thin rod (like a ski). To a skier who is interested, all they really should get out of this formula is that if you increase the length of a ski by 10%, then your ability to twist the ski in air goes down by a lot because the mass has increased by somewhere around 10% AND the radius of gyration has probably also increased by about the same amount, etc. etc.

If you know what you are doing, use the formulas for trends, not for details. If you aren't familiar with using formulas, don't worry about it, they are not that important from the point of view of an end user.

I know it may sound like I'm finking out on all the other geeks of the world, but I come up with the same bottom line as Wink: Listen to other people, read mags, lurk the forumns, etc to come up with a short list for demo day, then try them yourself.

PM
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