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A bicycle is going down an infinite hill, is there a limit to how fast the wheels can turn and, hand in hand, how fast the bike can go? The bike cannot break.

Qmechanic
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bicyclequestion
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3 Answers3

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Bike can go no faster than speed of light. The rim of the wheel Lorentz – contracts as velocity of bike increases.

The rest length of the rim of the wheel must remain constant. This means that the rim Lorentz contracts, and that the radial extension of the wheel contracts accordingly. The result is that the wheel become infinitely small in the limit that the bike moves with the velocity of light.

If $v$ is velocity on the rim in the rest frame $K$ of the wheel, we have $\Omega=v/R$, where $R=R_0/\gamma$ is the contracted radius of the rotating wheel, and $R_0$ is their radius when they are at rest. The angular velocity of the rotating wheel is then

$\Omega = \gamma v /R_0$

Hence, in this case the angular velocity $\Omega$ must approach an infinitely great value in $K$ when the speed of the bike approaches that of light. As observed in the hill’s frame $K'$, the distance between the marks on the path each time a point on a rim of the wheel leaves it is

$l'=\gamma 2 \pi R = 2\pi R_0$

and this distance is independent of the speed of the bike, even if the radius of the wheel decreases with increasing velocity, because the distance between the marks depends upon the rest length of the rim of the wheel and not their Lorentz contracted length. Also in this frame the angular velocity of the wheel remains finite even if the wheel have a vanishing radius when the velocity of the bike approaches that of light,

$\Omega'=\gamma^{-1} \Omega = v/R_0$

and hence $\lim\limits_{v \to c} \Omega' =c/R_0$, which is finite.

Details in the "Relativistic Trolley Paradox" in the Am. J. Phys, June 2016. There are two resolutions with either Lorentz contracted radius of the wheel or with constant one.

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The limit in any real situation (other than the speed of light) would be the tensile stress in the wheel. Due to centrifugal force the spinning wheel wants to fly apart and is only held together by the strength of the rim.

stress = density * radius^2 * angular velocity^2

So a typical bike wheel with 700mm diameter (0.35m radius) and assumign it was made of pure graphene (the strongest natural material?) with an ultimate tensile stress of 130GPa. Don't know the density of graphene, so will assume = 1

angular velocity = sqrt( 130GPa / 0.35m^2 ) = sqrt(1E12) = 1E6 rad/s = 165,000 rev/s

With a circumference of 2m this is 165,000*2 = 300,000 m/s or 0.1% the speed of light

(edit: sorry on a screen will latex it later)

Martin Beckett
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  • Remember the spokes... – Floris Mar 28 '17 at 22:28
  • @floris, have a vague memory that a fly wheel is limited by the hoop stress. – Martin Beckett Mar 28 '17 at 22:36
  • Yes - but a bike wheel is typically a rim and spokes. And the spokes provide an inward force which must relieve the tension on the rim, surely. But maybe the mass they add cancels any benefit. – Floris Mar 28 '17 at 23:05
  • @floris, I meant that I think the tensile stress in the rim is always larger than the stress in the spoke - at least for a flywheel – Martin Beckett Mar 28 '17 at 23:18
  • Since the question mentions an infinite hill, this suggests that the bike is just pulled by gravity. I believe in that case, air resistance/drag will limit the velocity much before 0.1% of the speed of light. – user1583209 Mar 29 '17 at 06:53
  • @user1583209 as a physicist I naturally assumed a spherical bicycle in a vacuum. Although I would also assume a frictionless slope - in which case the wheel wouldn't rotate ! – Martin Beckett Mar 29 '17 at 17:42
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Wind resistance is a major factor in spinning spoked wheels of a bicycle , most of the resistance you feel attempting to pedal at faster speeds mostly wind resistance you your self cause and the wind resistance through the spokes. It become progressively worst really quik . An engine connected to a 26 inch spoked wheel solo on a stand without a tire over comes the engines ability at a certain speed far from centrifugal destruction. I thought I add that in .

Russ Russer
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