Buying a bicycle - what wheel size do I need? is a very good read. It essentially teaches you that the larger the wheel the harder the effort.
Just jump on a kids bike with small wheels and see how easy it is to ride, but each revolution of the pedal gives you a shorter distance. Basically, without gears we would need bicycles of different wheel sizes for different rides. Thanks to the invention of gears.
Gears essentially give you many wheel sizes. If you take the chain ring (front gears that the pedals are attached) and divide it by the amount of teeth in the rear cog (that's the gear on the rear wheel) and multiply the result by the wheel diameter you will get the inches (meter, millimeters) you travel in one revolution of the pedal.
Suppose you have a road bike with 3 front gears and 10 rear gears, that essentially gives you 30 gear combinations and many wheel sizes. If one front gear has say 54 teeth and a rear gear has 18 teeth, then 54/18 gives you a gear ratio of 3. If your bicycle has a 26 inch diameter wheel, then one pedal stroke will drive you 78 inches forward (3 x 26). Remembering that Pi or π is one of the most important mathematical constants, approximately equal to 3.14159 (22/7) which times the diameter gives circumference. With this basic formula we can calculate different wheel combinations to gear ratios. Thus, by using the 54:18 gear combination, we get a wheel size of 24.8 inch diameter although the bicycle is equipped with 26 inch wheels. On the other hand, a 54:10 gear combination would give a distance travel of 140.4 inches per pedal stroke and a wheel diameter of 44.7 inches. The 54:18 ratio takes you a shorter distance (78 inches), but it is an easier ride. The 54:10 gear combination gives a travel distance of 140.4 inches but it is a harder ride. That is why you use lower gear ratios on uphills and on rough terrain (for stability). Imagine how painful if would have been if we had to carry different wheel sizes for a long bicycle ride. Thanks to the science of gears, we can get different wheel sizes as we change gears, making the ride easier when we need it and giving higher speeds (higher gear ratios) on the flats.
For mountain biking, a 26 inch wheel diameter is best and for road bikes 27-28 inches (700mm).
Applications in Bicycle Frames is a tad too technical, however, it is as scientific as it can get when you are looking for a frame that gives a comfortable ride. Just throwing words like carbon, titanium, steel and aluminum may be nothing unless the materials lend themselves to production designs than the material itself. Most bicycles, whether in aluminum or other metals will last for many years, unless you are a professional rider covering a thousand or so miles a month.
Seat and position of handlebars also add to a comfortable ride. For road bikers, the drop bars are efficient aerodynamically, but if the drop is too much and the stretch (seat to bars) too long, the ride can be uncomfortable and risky. For trails and mountain-biking, the flat bars offer stability. Also, for commuting, flat bars give the rider a better view of traffic and other surroundings.
The main thing is to get a good fitting bicycle that is appropriate for your use, racing, mountain-biking or plain leisure riding. If you use a bicycle for commuting, unless the bicycle is safely parked and secure from theft, you wouldn't want to purchase an expensive bicycle for that purpose.
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