Answer
$F=Mgtan(\frac{\theta}{2})$
Work Step by Step
We call the center of the wheel the axis of rotation. We know that the wheel will not rotate when the net torques are zero. We know that the torque due to friction is given by:
$\tau = r Mg sin(.5\theta) cos(.5\theta)$
Where r is the radius of the wheel.
We know that the force on the wheel is given by: $ Fcos^2(.5\theta)$
Setting these two expressions equal and solving for F gives:
$F=Mgtan(\frac{\theta}{2})$
