Analyzing a wheel rolling on a curved surface
When a wheel rolls without slipping on a curved surface, its dynamics can be understood by analyzing the velocity and acceleration of key points, such as the center and the contact point. Using normal and tangential coordinates simplifies the process and offers a clear representation of the motion.
The velocity of the wheel’s center can be derived using the relative velocity equation. Taking the instantaneous point of contact as a reference, I find:
\mathbf{v}_C = r \omega \mathbf{e}_t
This result shows that the velocity is tangential to the curved path, with magnitude proportional to the angular velocity \omega and the wheel’s radius r.
Acceleration analysis provides further insights. Differentiating the velocity yields two components: a tangential acceleration and a normal acceleration:
\mathbf{a}_C = r \alpha \mathbf{e}_t + \frac{r \omega^2}{\rho} \mathbf{e}_n
Here, \alpha represents the angular acceleration, \rho is the path’s radius of curvature, and \omega^2 contributes to the centripetal acceleration.
The acceleration at the contact point is determined using the relative acceleration equation. After simplifications, I obtain:
\mathbf{a}_P = \left( 1 + \frac{r}{\rho} \right) r \omega^2 \mathbf{e}_n
This expression shows the combined effects of path curvature and rotational motion at the contact point. As the radius of curvature \rho \to \infty, the curved surface dynamics approach those of a flat surface.
For more insights into this topic, you can find the details here.