Chapter 7, Problem 7.1P

To determine

The generalized coordinates and description about the rolling disk.

Answer to Problem 7.1P

The generalized coordinates and description about the rolling disk are given.

Explanation of Solution

In order to describe the rolling of disk without slipping, four generalized coordinates are required. They are – x coordinate, y coordinate, angle $\theta$ corresponding to rolling and angle $\varphi$ corresponding to spinning. These are represented in the figure below.

Equation of motion for the rolling disk is,

    $\left(\cos \varphi \right)dx+\left(\sin \varphi \right)dy=Rd\theta$

Here, $\tan \varphi$ is $\frac{dy}{dx}$.

These equations are not integrable. Moreover, all the generalized coordinates cannot be combined into a single equation. Hence, the constraints are nonholonomic.

Conclusion:

Therefore, generalized coordinates and description about the rolling disk are given.