Sliding Rod with Friction Discontinuity

A uniform, rigid rod of mass $m$ and length $L$ slides on a flat, rough surface with two different coefficients of friction: $\mu_1$ for $y \geq 0$ and $\mu_2$ for $y < 0$ in the $xy$ -plane.

The rod is supported by two tiny bumps at either end, and the surface normal reaction force is evenly distributed between the two end bumps at all times. The friction force at each end of the rod is directed opposite to the instantaneous velocity at that end. There is an ambient gravitational acceleration $g$ (into the page).

Initially, the rod's center of mass is located at $(x_{\small{C_0}},y_{\small{C_0}}),$ and the center of mass is moving with velocity $(\dot{x}_{\small{C_0}},\dot{y}_{\small{C_0}} ).$ At this time, the rod is not rotating, and its length is perpendicular to the boundary between the two friction regions.

When the rod comes to rest, what is the $y$ -coordinate of its center of mass?

Details and Assumptions:

• The rod does not roll, and both ends always remain in contact with the surface (with weight evenly distributed between the two end points).
• $\mu_1 = 0.6$ and $\mu_2 = 0.3$ .
• $m = 1 \, \text{kg}$ .
• $L = 1 \, \text{m}$ ( $\text{m}$ stands for "meter" here).
• $g = 10 \, \text{m/s}^2$ .
• $(x_{\small{C_0}},y_{\small{C_0}} ) = (0 \, \text{m},0 \, \text{m})$ .
• $(\dot{x}_{\small{C_0}}, \dot{y}_{\small{C_0}} ) = (3 \, \text{m/s},0 \, \text{m/s})$ .
• You will likely need to use numerical integration to solve this problem.

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