Circumference and Pi

Here is a slightly modified problem from the 2015 AMC:

A clock with diameter 4x and a disk with diameter 2x are tangent at the 12 o' clock point on the clock face. An arrow on the disk points vertically upward in its starting position.

Suppose the disk rolls counterclockwise around the clock face. At some point, the arrow on the disk will point upwards again.

What time on the clock face will the disk be tangent to at this point?