Mass m attached to wheel rolling on a surface (no slip).
A mass m is attached to the perimeter of wheel of radius b. The wheel is massless, except for a point mass M at its center. The radius of the wheel is b.
The task is to find the velocity of the point mass as a function of $\theta$ and $\dot{\theta}$
The solution sheet gives this
$v^2=2b^2\dot{\theta}^2[1-cos(\theta)]$
As well as giving the initial positions
$x = b\theta - bsin(\theta)$
$y = b - b\cos(\theta)$
What I am struggling to understand is how they find the $x$ coordinate?
The solution sheet assumes that the origin is at the centre of the circle at the instant when $m$ is directly below $M$.
When the wheel has rotated an angle $\theta$ without slipping, the point of contact of the circle with the floor is at a horizontal distance from the origin equal to the arc length between $m$ and the point of contact at this instant. This distance is $b\theta$.
Therefore, the horizontal displacement of $m$ from the origin is $$b\theta-b\sin\theta$$