Armstrong had a fast cadence, and he won a millon Tours de France, so we should all ride at a high cadence, right?
If you ask 10 cyclists about the importance of cadence, you'll get 3 different answers and 7 blank looks. Cadence is confusing, but the basic fact is that riding at a higher cadence is faster, except when it isn't.
High Cadence is Faster
So, you went out on that hilly century to ride with some friends. You felt good and fast on the hills, but by mile 50 your legs were burning, and you could barely make it up the later hills.
Power is the product of force - how hard you push on the pedals - and cadence - how fast your turn the pedals. Drop down a gear or two, push 20% easier and spin 20% faster, and you get up the hill at the same speed.
But you save your legs. If you've ever weightlifted, you know that a 20% difference in weight can make a huge difference in how many repetitions you can complete. The same effect is at work on your bike - spin instead of mashing, and you can climb 7 hills before your legs give out instead of 5.
It's also true that riding at a higher cadence helps you develop a better pedal stroke, which recruits more muscles and uses them over a longer period of rotation, lessening the peak force for a given amount of power.
And it helps you accelerate faster when you need it.
Higher Cadence is Slower
When you spin, you put more load on your heart and less on your muscles. All things being equal, for a given amount of power, spinning faster will take more aerobic capacity, so you'll be more out of breath on that climb.
And, if you spin all the time, you can develop a great aerobic system, but you don't stress your leg muscles, and they don't get stronger.
A Proper Balance
For a given individual in a specific state of training and a particular ride, there's a "fastest cadence". It's the one where you use up your leg strength right as you finish the race. Ths could easily be faster than your current cadence, and it's therefore worthwhile spending some time working on cadence.
My Cadence Story
When I got serious at cycling a few years ago, my average cadence was in the 80s, and I decided to work on getting better at it. By just trying to spin faster, I got myself up to a top cadence of around 100RPM, but I wasn't very comfortable of it.
Then when I did the Carmichael training a couple of years ago, they specified specific cadence drills to do. I did them that summer, and over the past few seasons. My max on-bike cadence is now up into the mid 140s, and I can high 120+ pretty easily. My average is a little bit higher, but not a ton higher, partly because I spend a lot of time on low-cadence muscle tension work on climbs.
On a flat or slight uphill with medium resistance, slowly increase your cadence over a period of 30 seconds until you hit your maximum comfortable cadence. Hold that for 30 seconds, and then slow down over the next 30 seconds. Slow down a bit if you start to bounce. Do 3-4 repetitions of this.
Initially, two things will happen. First of all, you're going to feel a bit out of control. That's because your muscles aren't used to spinning that fast and you need to "rewire" your brain to make them work at that speed. Second, you are going to get really out of breath because you aren't very efficient at that speed yet.
Once that starts to feel a little more natural, increase the time at which you hold the top cadence to 60 seconds. Do this once or perhaps twice a week, but you don't really need to overdo it.
I also find it useful to do one-legged pedalling drills, where you clip out one side and do a fixed number of revs (20-30 is a decent place to start) on one foot, then that same number on both feet, and then the other foot, etc. Do this one the flat and in a quiet area, or do it on a trainer, as they feel really weird.
I can hit the low 120s fairly easily now on the bike, 130 with some effort, and I saw 140 with a slight downhill a while back (though I was bouncing a bit at that speed). On the recumbant exercise bike at the club where I work out I've hit 160 for short periods.