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I am looking for a curve traced by a moving bicycle when its steering wheel is fully rotated either clockwise or anti-clockwise.

How to model it mathematically?

Is the curve a circle?

My attempt is as follows:

Let $\vec{r}_1(t)$ and $\vec{r}_2(t)$ be the position vectors for the tangent points (between road and tires) on the rear and front tires, respectively. I know that $|\vec{r}_2(t)-\vec{r}_1(t)|$, $|\dot{\vec{r}}_1(t)|=|\dot{\vec{r}}_2(t)|$ and $\dot{\vec{r}}_1(t)\cdot\dot{\vec{r}}_2(t)$ are constants. $\dot{\vec{r}}_1(t)$ is in the direction of $\vec{r}_2(t)-\vec{r}_1(t)$.

Assuming the tires rolls without slipping then their linear velocity is the same.

  • 0
    I see that $|\vec{r}_2-\vec{r}_1|$ is the fixed distance between the front and rear tires, but why does $|\dot{\vec{r}}_2|=|\dot{\vec{r}}_1|$?2012-07-30
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    @robjohn: rolling without slipping.2012-07-30
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    However, if $|\vec{r}_1|\not=|\vec{r}_2|$, they can have the same angular velocity, but different velocities, or am I missing your point?2012-07-30
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    @robjohn: If the rear tires rotate and travel $\Delta s$ unit on the road then its center moves $\Delta s$ and the front tire's center also moves $\Delta s$.2012-07-30
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    What you say is true if the tires are parallel. If the tires are not parallel, then the car turns and the tires follow concentric circles as shown in [my answer](http://math.stackexchange.com/a/176898). This is why the back tires of a car or bus always make a tighter turn than the front tires.2012-07-30
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    @robjohn: In my mental model, there is only one circle for both front and rear tires.2012-07-30
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    It is the same for a bicycle. When turning, the front tire moves faster, and in a larger circle, than the back tire.2012-07-30
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    The only way that both could follow the same circle is if both tires were at the same angle with respect to the body. Each tire moves perpendicular to its axis.2012-07-30

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