A disk with radius r rolls without slipping on a stationary

A disk with radius r rolls without slipping on a stationary surface described by a circular arc of radius . For cases (a) and (b), derive the acceleration of the contact point between the disk and surface in terms of the given quantities. What happens to these accelerations as approaches infinity and what does this mean?

This is a homework question in my Dynamics class. I would appreciate any help, thanks!

A disk with radius r rolls without slipping on a stationary surface described by a circular are of radius rho. For cases (a) and (b), derive the acceleration of the contact point between the disk and surface in terms of the given quantities. What happens to these accelerations as rho approaches infinity and what does this mean?

Solution

For both the cases, there will be both tangential and centripetal accelerations. Centripetal acceleration will be directed from the contact point of the disc with the surface towards the centre of the curved surface. It is given by

  a_c = * ² towards the centre from the point of contact

Tangential acceleration will be along the tangent and in the same sense of angular acceleration , i.e. towards right. It is given by

a_t = * towards right at the point of contact along the tangent

The total acceleration of the disc is a_T = {a_c² + a_t²}

When the radius approaches infinity, the centripetal acceleration will tend to become zero. This is obvious because when the radius approaches infinity, the surface tends to become a horizontal surface and on this surface there will not be any radial acceleration.