Serious cyclists (and some who aren't so serious) obsess over every ounce on their bike. Yes, a lighter bike can save you some energy. And a rule of thumb states that mass on the wheel is like twice the mass on the frame.
But why? Let's look at some possible effects of extra mass on a bike.
To ride at a constant velocity, the net force must be zero. Of course there is the gravitational force pulling down and the road pushing up. But there also is air drag force pushing against the direction of motion. You feel this same when you stick your hand out of a car window. The faster you go, the greater the force.
What happens to air drag force if I add mass to my bike? If the added mass doesn't change the shape or cross sectional area of the bike, the force remains the same. Air drag force doesn't depend upon the mass of the object. But what if the extra mass pushes down on the tires, causing the tires to stick out more and cause more air resistance? OK, that's technically possible—but it wouldn't matter if the mass was on the bike frame or the wheel.
Actually, in terms of air drag, extra mass can help. Suppose you have two bikes that look the same but have different masses. If they are traveling the same speed, they will have the same air resistance force on them. However, this force will produce a greater change in speed on the bike with less mass. Don't forget that a net force is equal to the produce of mass and acceleration. Same force but different masses means different accelerations.
If you get a bike going and coast, the bike slows and then stops. This would occur even without air resistance. As a wheel rests, the part of the tire touching the ground compresses and deforms. When the wheel turns, the section of tire being compressed changes. The constant compression and relaxation requires energy. This is called rolling friction.
What if you put a mass on the frame? The tire is compressed further, resulting in greater rolling friction. How about if you put the mass on the tire, not the frame? The same thing happens, but you could argue that the effect isn't as great. If the mass is evenly distributed around the circumference of the wheel, then a part of this mass will be at the contact point and not really push down on the bike. This might be true, but the effect would be tiny.
When you pedal, you're working against friction in the wheel bearings, friction in the bottom bracket, friction in the chain and through the cogs and chainrings. This reduces your efficiency. But what about adding mass? More mass on a bike can increase the friction in the bearings---but again, this won't matter where the mass is located.
If you have a book on the floor, you could pick it up and put it on the table. However, since you have to push on this book as you lift it, you will do work on the book. You could also say that increasing the height of the book changes its gravitational potential energy. On the surface of Earth we can define this potential as:
The greater the mass, the greater the change in potential energy. Where does this energy come from? It comes from the rider. So again, more mass means more work for the human. But still, it doesn't matter if the mass is on the frame or the wheel. You still have to increase its potential energy.
An increase in speed means an increase in kinetic energy. Since the kinetic energy depends on both mass and velocity, more mass would mean more energy required to speed up.
But does it matter where this mass is located? Does it take more energy to increase speed if you put the mass on the wheel? Yes. First, let's look at mass on the frame of the bike. If I add something to the frame the total mass increases. This means that I would need more work to increase the kinetic energy. That's pretty straight forward.
What if the extra mass is on the wheel? In that case, I must do two things to increase speed: increase the kinetic energy and increase the rotational kinetic energy of the wheel. If all of the mass on the wheel is located at the rim, I can write the rotational kinetic energy as:
In this expression, mw is the mass of the wheel, R is the radius of the wheel and ω is the angular velocity of the wheel. But if the wheel is rolling and not slipping then there is a relationship between the angular speed of the wheel and the linear speed of the bike (this is how a car speedometer works---or at least the way it used to work).
If I substitute in for ω, I can write the following for the total kinetic energy of the bike (translational plus rotational).
In the translational kinetic energy, mb is the total mass of the bike (including the wheels) but the rotational kinetic energy only depends on the mass of the wheels.
So let's say I add 100 grams to the frame. This would increase the value of mb but not increase the mass of the wheel. The translational kinetic energy would increase by some amount and it would require more energy to accelerate (increase the kinetic energy).
Now let's add 100 grams to the wheel (increasing mw). Since the wheel is part of the bike, this means that the total mass also increases (mb). Both translational and rotational kinetic energy terms will have a 100 gram increase in mass. You will have double the increase in energy by adding mass to the wheel.
So yes, adding mass to the wheel is worse than adding mass to the frame---but only when accelerating. Still, every little bit helps.
