Relationship of weight and effort?

Why do we attempt to reduce the weight of our packs? Another way of saying that is, what is the price of additional weight? It is a rhetorical question, but a short list might be:
1. More work to go the same distance (work in joules, ergs, foot pounds, etc.)
2. More displeasure (less fun) with more weight
3. Pain (sore back, muscles, chafing.
4. Less distance possible.
5. Less speed possible.
6. Elevation gain more difficult
Now suppose you sum up all those things as some quanity. Call it effort. That quanity is a function of weight. That is to say more weight equals more effort. But it isn't a simple linear relationship. It seems to me that carrying a 50 pound pack is more than twice as difficult as carrying a 25 pound pack and at some point it becomes quite impossible. So I strongly suspect the “effort” is increases quickly with the back weight.
So we have a quanity we want to model with respect to weight. Effort with respect to weight. Perhaps it could be expressed like this: Effort is a function of the weight to some exponent. If y = effort and x is weight then perhaps y=x^n and we can throw in a constant. y =ax^n. Or written a slightly different way f(x) =ax^n.
If n =2 then if you compare say 20 pounds with 40 pounds it isn't twice the effort but rather four times the effort.
Does quantifying the relationship like that make sense to anyone?

History major here, so I have no idea if the math is right. However, I think you can apply the same logic to any extra body weight you carry. I know that, with the same pack weight, I walk more slowly and end the day more tired when I’m 10 pounds overweight than when I’m not. (Sadly, I’m over more often than not. 😔)

There may be some underlying linear relationship but I have always had a break-point over which the effort and discomfort become significant and quickly get into the miserable to impossible. Sort of like climate change "tipping point". When I carry a heavier pack it is usually for a longer duration trip, like two weeks. Then I do get close to my limit.

Much more complex than a simple formula. The comfort part of it is dependent on your pack fit and suspension and how you pack it. Also lots of walking techniques involved, such as rhythmic breathing or the rest step that allows you to carry heavier loads uphill. Also just plain mental; get in the right state of mind and you can do amazing things. I find unexpected difficulty more difficult than expected difficulty!

In addition to "costs" some weight has "benefits" that outweigh the costs. Not everyone wants to go faster or farther. Depends on your backpack style. Some are willing to "pay for" camp convenience or comfort or the ability to partake in an activity (hobby).

IMO it will prove to be a hyperbolic curve.


More of a slope when I was young and dumb, but now that I'm much older and still dumb, find I can hit the oh-no-you-don't weight wall much lower on the scale. And yet a, say, 10-pound daypack seems no different than walking.

An additional factor is age...and in my case arthritis in my knee. Weight seems more important today than it did forty years ago...go figure.

For quantification of the pure energy cost of additional weight, I believe that effort is directly proportional to total weight of the loaded hiker, at least for loads that aren't too large (whatever that means).

I'm pretty geeky even for an engineer so tested this hypothesis a while ago with a light running pack and a heart rate monitor. I jogged the same short loop four times in succession, alternating between not carrying and carrying a small pack weighing 10% of my body weight. I don't have the numbers any more but the indicated calories for the runs were something like 200,225,210,235. This suggested to me a 10% cost for the 10% extra weight coupled with a fatigue factor indicating decreasing efficiency with time. A more applicable test for hikers would be to do one walk per day with or without a back and maybe use a more realistic backpacking load at hiking speed.

Physics-wise, walking and running are both controlled falling: you lift your body+load a few inches, lean forward, and let yourself fall back down that few inches while you move a foot forward and catch yourself. The energy cost to lift a known mass against gravity is pretty straightforward and dominates the total energy cost when using a normal gait. I believe that the DIFFERENTIAL energy cost of carrying additional weight has very little dependence on speed or incline within some "reasonable" range. This simplification probably breaks down with heavy loads, though, because there's additional cost in controlling the "fall" phase and excess weight throws off biomechanical efficiency.

As someone else pointed out, in the real world things like knee pain are the real limiting factors for big loads. Still, I bet that the weight percentage effort simplification is pretty accurate in predicting, say, the cost in miles/day of carrying a few extra pounds of gear over the same route. By this reasoning, a 100-pound hiker that's used to carrying a 20-pound pack 20 miles/day would only make it 18 miles with the same effort if she had to carry an extra 12 pounds of food. Alternately, she'd cover 24 files if she slack-packed. It also means that that 2-pound luxury item you're considering costs about a third of a mile or about 10 minutes of hiking when you think of it in terms of a percentage of body+pack weight.

Thoughts?

I'm surprised by your numbers. Not arguing or disagreeing, just surprised.

Many years ago I belonged to a gym with a variety of exercise machines that, among other thing, displayed calories burned. I also looked up the number of calories burned by activity on the internet (as we know everything on the internet is true). I performed the various exercises over several weeks and gauged how I felt. So, very subjective.

I determined that carrying 1% extra weight burned 5% more calories. So if one burned 100 calories per miles, a figure often cited for the average male, then carrying 10% of one's bodyweight should burn an extra 50% calories, so 150% of the baseline or 150 calories per mile. Carrying 20% of one's bodyweight, about what I carry for a backpacking trip, should burn 200 calories per mile. My more recent use of apps confirms this.

Using these figures, if one hikes 10 miles then one burns about 2000 calories. Add this to a baseline (for me) of around 2400 calories per day and one needs around 4400 calories per day to backpack 10 miles per day. Based on what I've read of thru-hikers (of which I am not, yet) this seems to be in the ballpark.

So those are my thoughts

Totally unscientific here: I do not think you can exactly extrapolate from running to walking. And I do not think calories used is a good measure of "effort". What is left out is the aerobic "cost". Running you are already at close to your aerobic maximum, whereas walking you have a lot of room to push more aerobic demands. "Effort" is also tied to how you feel and at what point you start to hurt and psychologic burn out as well as physical burn out (hitting the wall). These are very subjective but real. Altitude and elevation gain are also left out of these calculations. As is overall fitness and getting used to a heavier pack the more you do. An added 5 pounds at the beginning of the backpack season takes much more effort to carry than the same at the end of the backpack season. I think everyone has, through experience, their own unique feel for the extra effort a heavier pack requires. Too many variables to put in a universal equation.

I wear a Garmin watch (which as been reviewed as one of the more accurate fitness watches) that estimates calories used based on "measured" heartbeat, and I find it often inaccurate. According to my calories burned, given what I actually eat I should be obese by now!

As with most things, reality is probably somewhere in between. A 5% energy expenditure increase for a 1% weight increase seems high: if that were true I think it would be a lot easier to lose body fat. I'm about 10% higher than my "racing weight" of a few years ago but I know that I'm not eating 50% more calories just to maintain equilibrium. If you really want to geek out on this I found a paper describing metabolic cost of weight on ankles, knees, and back that puts numbers to the well-known extra penalty of boot weight:

http://www.bgu.ac.il/~rriemer/papers/Schetzer%20Riemer%202014%20Metabolic%20rate%20of%20carrying%20added%20mass%20A%20function%20of%20walking%20speed,%20carried%20mass%20and%20mass%20location4.pdf

They seem to confirm prior work indicating a linear relationship for weight held close to the core and increased penalty at the extremities. They also measured O2 consumption with a mask, which has to be much better than my cheap HRM.

Thank you Rick, yes I'm sure there is a Log in there, probably an inverse Log. HOWEVER its too simple of a justification to spend money on new gear.

You must consider the weight and condition of the person carrying said pack. The pack itself does not get up and walk thus its actual "weight" can be only be thought of as a part of a whole. Regardless of the weight of my pack, there are atleast 5 maybe 8 pounds of gear on my body besides my pack and food and water.

Note the water part. While we sit in our warm Winter chairs and bravely write about our camping prowess we forget some small things like trying to light a cigarette in a blizzard and the real weight of water. If you hike where I do, it takes more than a quart of water for a days hike. The weight of water quickly adds up, like food, but we don't drop food to make up for water, instead we are happy to pick up 4 or even 6 pounds of water and add it to our weight because it isn't dead weight that must be carried back home.

I had a backpack girl friend who would bring half a duck, bottle of wine, premade salads etc. We might actually carry in two 40 pound packs, but by the time we left, they were about 25 or less. I know it sounds crazy but she was a very entitled very rich girl.

So back to saving one pound or even two by spending $500. Its Lame If you are in fact a genuine high altitude mountaineer or something then get the best, but here's a thought:

You can pick up everything actually needed to camp at a thrift store for $50. Assuming no extremes in weather. I spent a summer once hitch hiking with little more than sleeping bag and a pack of matches. I remember getting a cheap backpack and then a few years later someone invented the waist belt Then I got a blue closed cell foam pad and I thought that camping had finally become a gentile sport.
Jim

Sounds Good Jim.
"I had a backpack girl friend who would bring half a duck, bottle of wine, premade salads" Do you still have her number?
the other Jim

41234
I like your ideas. Of course in physics I learned to always qualify my modeling with, "all other things being equal." Where we hike elevation gain is much more the culprit in my opinion. I can hike all day with an UL pack, but if I have to gain 3 or 4 thousand feet I am certainly bushed. Thanks for your comments. When I was young I often ignored extra weight or just grunted it out. No more.

41232
I did a couple of spreadsheets looking at how % of increase in body and pack weight might affect distance if they were inversely proportional. 10% more tot wt. = 10% less miles.
It didn't seem to be enough reduction in miles for me.
So I ran a scenario where 10% more PACK wt. = 10% reduction in miles.
That gave me half the distance if I increased pack from 18 to 36 lbs. (of course). So there seems to be more variables involved and I can't do calculus with more than two variables because I have an UL brain.

I used to be able to do multi-variable calculus. That convinced me to become a history major. 😏