Hi....
You are familiar with Lance Armstrong I take it right? So a few years ago he did a blazingly fast time up in a Time Trial on a very famous and difficult mountain stage in the Tour de France, Alpe d'Huez. If you look at the watts he produced, take into account his cadence, then the 'force' he was putting out with both of his legs was equal to about 55 pounds. There are few adults I know of, who can't do 55 pounds with 2 legs, I mean just climbing stairs requires that much....and more.
Lance reportedly rode up that mountain at around 475 watts or so. You know what though? I am certain you could do 475 watts - just not for an hour or anything close to it obviously, and neither can I!
Also, consider when you sprint you are exerting far more force than on any long climb, so you already have the 'strength' to do those climbs. The reason you, me, or anyone else typically struggles on a climb is because our bodies cannot meet the demands of our muscles - delivering oxygen, fuel mix, etc...and why? Our lactate thresholds, Vo2max, etc aren't developed enough for what we are trying to do.
In addition, weight training doesn't nearly replicate the joint angles and velocities of cycling. That's along the lines of the whole specificity principle, so those gains are not going to be very transferrable.
I'm not telling you weight training is bad or that you shouldn't do it - there are lots of benefits to it.
What I do think is that it won't make you a better climber on a bike. It can make you a better track sprinter or pursuiter - though if you gain weight (from muscle gains), it can then cost you in climbing and other aspects of riding.




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