Solution

We denote the radius of the circle along which the coil rolls around the instantaneous axis by \\$ R^* \\$:

\\$\\$
R^* = R \\\\cos\\\\alpha - r
\\$\\$

The angular velocity of the coil \\$ \\\\omega\\$ is defined as the ratio of the linear velocity to the radius:

\\$\\$
\\\\omega = \\\\frac{v}{R \\\\cos\\\\alpha - r}
\\$\\$

The velocity of the axis \\$ v_0 \\$ can be found by considering the rotation of the coil as a movement around the center \\$ O \\$:

\\$\\$
v_0 = \\\\omega R
\\$\\$

We substitute the found angular velocity:

\\$\\$
v_0 = \\\\frac{v R}{R \\\\cos\\\\alpha - r}
\\$\\$

The coil will move to the right if the velocity \\$ v_0 \\$ is positive, and to the left if \\$ v_0 \\$ is negative. The condition for changing direction is:

\\$\\$
R \\\\cos\\\\alpha = r
\\$\\$

Accordingly, the critical angle \\$ \\\\alpha^* \\$ is determined from:

\\$\\$
\\\\cos\\\\alpha^* = \\\\frac{r}{R}
\\$\\$