Bicycle gears paradox

My wife and I got new bikes recently, and she’s still trying to get used to the new shifters. Our old bikes had the grip shifters with the numbers that told you exactly what gear you were in. The new bikes have the shifters that are integrated into the brake handles and there is no visual cue as to what gear you are actually in.

So the last time we were out on our bikes, she was getting very frustrated and confused trying to get the bike in the correct gear just by feel. I tried to explain to her that pushing the big shifter puts the chain on a bigger gear, and pushing the smaller shifter puts the chain on a smaller gear.

But where it gets confusing is, with the front set of gears, the *bigger * gears are harder to pedal, whereas on the rear, the *smaller * gears are harder to pedal. Why is the behavior different for the front set of gears than for the back? I’m sure there is a simple answer, but I’m not seeing it.

The mistake you are making is that you are not taking into account is that the chainwheel is the driving gear, the larger it is, the harder to pedal, and the sprocket is the driven gear, so it works the opposite way around, the smaller it is the harder to pedal and vice versa.

You can get an idea of gear ratios by looking at the math.

Chainwheel teeth = a

Sprocket teeth = b

ratio = a/b (multiplied by the circumferance of the cycle wheel)

so you can see, the smaller b is, the higher the gear ratio will be

the larger a is, the higher the gear ratio will be.

Bike gears work by adjusting tension on a cable against a spring in the gear mechanism. Increasing the tension pushes the chain onto a bigger sprocket and releasing it allows the spring to drag the chain back to a smaller one. The apparent ‘paradox’ is because, on the rear mechanism, the smaller the sprocket the higher the gear whereas on the front the smaller chain ring the lower the gear. Thus the levers, whilst doing the same job of tensioning/releasing the cable, have an opposite affect.

It’s like using a lever to pry something. To get maximum force at the business end, you want that end to be short and the handle end to be long. The point is that more length has opposite effects on the two ends.

Think of it like this:

The chain has to drive the back wheel, which pushes the bike. Look at it like a pully. The wheel is a fixed size – say 27". Let’s say the large gear is 10" in circumferance, and the small gear is 5" in circumference. When the chain is driving the large (“easy”) gear, then, you’re using 10" of chain for one wheel revolution. On the small (“hard”) gear, you’re driving 5" of chain. In either case, the work is the same, but you need different efforts for differing amounts of times.

Now what’s counter-intuitive is the front sets of gears. The big gear is the “hard” gear and the small gear is the “easy” gear. At this point, you are driving the chain. Is it easier for you to drive 10" of chain per leg revolution, or 5" of chain per leg revolution. Alternatively, which can you do faster given the same amount of force applied? With the small gear, you’re able to drive it with less effort, but you get less work. And of course with the big gear, you’re moving even more chain. You need more effort, or more time. If you do it in the “same” time, then you’re moving faster, but at the cost of more energy.

But is there a reason it is done this way? Could you have the biggest gear on the inside for the front crank? If not, could you have the cables strung so down is higher gear for both front and back?

I’m not particularly confused by the arrangement, I’m just curious if it is that way for physics or historical reasons.


The gears are aligned that way to provide the minimum sideways torque on the chain for the most common gear ratios. I ride most of the time in the top gear, which has the chain in a straight position. If I were to shift to the lowest gear on the back sprocket, that would put a tremendous stress on the chain and eventually wear it out where it would break. Same goes for the reverse – there’s no real good reason to ride with the front and back sprockets on their lowest gears – it’s more useful to ride with the front on the smallest and the rear on the largest (ie the very bottom gear of the bike), and this alignment is done to put the elast stress on the chain. On your typical 21-speed bike (3 on the front sprocket, 7 on the rear), there are really only about 12 or 15 useful gear ratios.

With bike gears, due to the tension from twisting the chain, you rarely pedal in smallest chain ring and largest rear gear or vice versa. Those two combinations are basically off limits, since they stress the chain too much. If you reversed the front chain ring you would eliminating your biggest and smallest combination, rather than two combos in the middle of the range that are essentially redundant.

You want your biggest and smallest combos, for going up and down steep hills.

One other thing tha tends to complicate things is the fact that on many shifters to make the pedaling easier you push the front shifter one way, but the rear shifter the other. This is the way my first hardtail, and my road bike are. Now to complicate things, my new full suspension has a reverse throw rear derailleur so it shifts backward from my other two bikes. :smack:
My suggestion is that you get on flat ground, put the chain in the middle chainring (assuming you have 3 chainrings), put the rear in about the middle. Ride on flat ground and practice shifting up and down both front and rear until you hands learn which lever to make the pedals easier, or harder.

Actually, this is what I suggested that my wife do next time we go riding. Use the middle gear in the front, and just practice moving up and down on the rear gearset. I think she’ll get the hang of it much faster if she concentrates on one shifter at a time.

And thanks all for the simple physics lesson!

Something that may help this make sense to you is expressing gear ratios in wheel inches, that is what would be the equivalent wheel diameter of an ordinary, a big wheel bike commonly called a penny-farthing. This made sense of the gears on new safety bicycles to customers used to ordinaries and is still in use today.

Most all terrain bikes have a nominal 26" wheel but let’s use 27" as is common for road bikes. Start with a middle gear of say a 48t chainring front and a 24t sprocket at the rear. For each revolution of the cranks the rear wheel will turn twice. Multiply that factor by the diameter and you have a so-called 54" gear.

Make the rear cog bigger, 32t and now the rear wheel turns 1.5 times for each turn of the cranks for a 40.5" gear. For each revolution you won’t go as far but it will be easier to push the cranks.

Make the front chanring bigger, 52t with the original 24t sprocket cog as we started with. That gives a 2.1666 ratio for a 58.5" final gear. You go farther for each revolution but it’s harder to push.

I am twenty-six years old and I’ve been a mountain-bike owner for aproximately sixteen of those years. In that entire time I knew that the front deralier functioned oppositely to the rear. In other words putting the chain on a bigger cog at the front made it harder to ride, but putting the chain on a bigger cog at the back made it easier. It’s simple physics. the front is the driver, the rear is the driven. The bigger the driver and the smaller the driven, the harder it will be to pedal (but the bike will go faster) and vice versa.

I’ve never owned a bike where I knew the number of the gear I was in. I always chose a gear which felt most appropriate for the incline, speed, intent to get where I was going I was in.

Let me restate what’s been said a bit differently.

Since all the gear teeth are equally spaced, the number of teeth is just a result of the size of the gear wheel, so let’s forget about the gear teeth for the moment. Just look at the length of the levers, where the fulcrum is, where the force is being applied, and where the output motion occurs.

When you’re pedaling, your foot is at one end of the lever. Obviously, you get the most mechanical advantage when the thing you’re trying to move (the chain) is nearest the fulcrum (crank axle). That’s on the smaller gear wheels. Therefore, in the front, the smaller gears are easier to pedal.

The rear wheel is the reverse. In this case, the force (chain on gear wheel) is now being applied closer to the fulcrum (rear axle) than the resultant motion (the outside of the wheel). Therefore, in the rear, the smaller gears are harder to pedal.

Coming late but here’s my take:

If you can agree that a gear (regardless of its design) is just a ratio between the angular speed of two wheels, the rest is simply basic maths. For example to decrease the ratio a:b you can either decrease a or increase b (and vice-versa).

Physics plays in when you include Work into the problem.