Physics · Mechanics · Grade 7

Circular motion: the bicycle

Why does a bicycle speed up when you change gear, though you pedal at the same rate?

The situation

You are exercising on a bicycle mounted on an indoor trainer: the rear wheel touches a roller and turns it, and the roller drives a heavy flywheel through a belt. You turn the pedals 60 times a minute, but if you change gear, the speedometer shows a completely different speed. With what speed and acceleration does a point on the tyre move? Why do drops from a wet wheel fly off along the tangent?

Topic

Uniform circular motion. Bicycle wheel and sprockets: angular velocity, period and chain drive

Key ideas

Angular velocity and period
Centripetal acceleration
Chain drive

Formulas

ω = \frac{Δ φ}{Δ t} = \frac{2 π}{T} = 2 π ν

Angular velocity (rad/s), period of rotation $T$ and frequency $ν = 1 / T$ .

v = ω R

The relation between linear and angular velocity: in one wheel $ω$ is the same everywhere, while $v$ is proportional to the radius.

a_{n} = \frac{v^{2}}{R} = ω^{2} R, a_{τ} = ε R, a = \sqrt{a_{n}^{2} + a_{τ}^{2}}

Normal (centripetal), tangential and total acceleration; $ε = Δ ω / Δ t$ is the angular acceleration.

i = \frac{ω_{2}}{ω_{1}} = \frac{z_{1}}{z_{2}} = \frac{r_{1}}{r_{2}}

Gear ratio: the number of teeth in chain and gear drives, the ratio of radii in belt and friction drives.

\vec{v} = \vec{ω} \times \vec{r}, \vec{ε} = \frac{d \vec{ω}}{d t}

Vector form: $\vec{ω}$ and $\vec{ε}$ point along the axis (right-hand screw rule).

Theory

What describes circular motion?

The period $T$ is the time of one full revolution, and the frequency $ν = 1 / T$ is the number of revolutions per second (Hz). Cyclists measure pedalling frequency as cadence, in revolutions per minute: 60 rpm = 1 Hz. Angular velocity $ω = Δ φ / Δ t$ is the angle through which the radius turns in one second (rad/s); one revolution is $2 π$ radians, so $ω = 2 π / T = 2 π ν$ . If $ω$ is constant, the motion is uniform circular motion.

Linear velocity along the tangent

All points of a wheel rotate with the same $ω$ , but in one revolution a point on the rim covers a distance $2 π R$ : $v = ω R$ . At every moment the velocity vector points along the tangent to the circle — a drop that comes off the tyre is thrown in exactly this direction, and then, under gravity, traces a parabola (range and height — see the “Basketball: the free throw” experiment). The trainer's speedometer shows $v = ω R$ : on the road the bicycle would travel at exactly this speed.

There is acceleration even if the speed is constant

In uniform rotation the direction of the velocity turns continuously, so a point receives a normal (centripetal) acceleration directed towards the centre: $a_{n} = v^{2} / R = ω^{2} R$ . At 36 km/h, for a point on the tyre of a 700C wheel this is ≈ 300 m/s² — 30 times the acceleration of free fall; it is provided by the rim and the spokes. In any curvilinear motion $a_{n} = v^{2} / ρ$ , where $ρ$ is the radius of curvature (for a circle $ρ = R$ ).

Non-uniform rotation: tangential acceleration

If you push the pedals harder, $ω$ increases — an angular acceleration $ε = Δ ω / Δ t$ appears. A point on the rim now also has a tangential acceleration $a_{τ} = ε R$ along the velocity; the total acceleration $a = \sqrt{a_{n}^{2} + a_{τ}^{2}}$ deviates from the radius by the angle $\tan β = a_{τ} / a_{n}$ . At low speed $\vec{a}$ points almost along the tangent, at high speed almost towards the centre — watch the arrows (the $a_{τ}$ arrow is magnified so that it can be seen, and the factor is written out). The vectors $\vec{ω}$ and $\vec{ε}$ point along the axle of the wheel.

Gears: chain, toothed, belt, friction

The chain passes over the teeth of the sprockets without slipping: in one second the same number of links passes over both sprockets, so $z_{1} n_{1} = z_{2} n_{2}$ and $i = z_{1} / z_{2}$ . For toothed wheels in direct mesh the ratio is the same, only the direction of rotation is reversed. In a belt drive the belt speed is the same on both pulleys: $ω_{1} r_{1} = ω_{2} r_{2}$ , and the direction is the same. In a friction drive (tyre → roller) two wheels touch through friction: the speeds are equal at the contact point and the direction is opposite. The overall ratio on the trainer is the product of the three ratios. If the large sprocket is at the front and the small one at the back, it is a “heavy” gear: the speed is high, but the pedals are hard to turn.

A rigid body: translation and rotation

On the trainer the wheel only rotates. On the road it simultaneously moves in translation (the centre moves with $v$ ) and rotates: the velocity of a point is ${\vec{v}}_{c e n t r e} + \vec{ω} \times \vec{r}$ . The point of the tyre touching the ground is momentarily at rest, the topmost point moves at $2 v$ , and a point on the rim traces a cycloid (the “Relative to the ground” button). Angular velocity is a quantity belonging to the whole body: it is the same for every point of the wheel (moment of inertia and angular momentum — see the “Dynamics of a rigid body” experiment). Model: the chain does not stretch, the roller does not slip, the cyclist sets the cadence himself, and the pedalling acceleration is constant.

Did you know?

The pitch of a bicycle chain is 1/2 inch, i.e. 12.7 mm: each tooth of the sprocket corresponds to one link of the chain, so a 52-tooth sprocket pulls 52 × 12.7 ≈ 660 mm of chain in one revolution.
Bicycle wheels are designated by the bead seat diameter of the rim according to the ISO 5775 (ETRTO) standard: on the “700C” wheel of road bicycles this diameter is 622 mm.

Inside the experiment

The experiment starts from a real-life situation: you predict the result, change the controls, run the experiment and draw a conclusion. Inside are missions, theory, a quiz and a guided tour in which the camera moves and explains the key points.

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