(a) 0.394
The static frictional force provides the centripetal force that keeps the button stuck on the rotating platform. So we can write
where the term on the left is the centripetal force, and the term on the right is the frictional force, and
m is the mass of the button
is the angular velocity of the platform
r is the distance of the button from the axis
is the coefficient of static friction
g = 9.8 m/s^2 is the acceleration of gravity
In this situation, we have:
r = 0.220 m is the distance of the button from the axis
is the angular velocity
Re-arranging the equation for , we find the coefficient of static friction:
(b) 0.098 m
In this case, we know that the angular velocity is
And we also know now the coefficient of static friction,
So we can now re-arrange the equation for r:
And substituting, we find the maximum distance at which the button can be placed without slipping: