Determining Maximum Angular Velocity and Slip Time for Object on Turntable

[Sakura91]

Homework Statement

An object is placed on a turntable with a radius of 21.5 cm. The object's mass is 67.5 g. The coefficient of static friction between the two is .32. It asks to find the maximum angular velocity the object can have while remaining stationary relative to the turntable in radians per second.

The second part says that the turntable starts from rest at t = 0, and it has a constant angular acceleration of 3.82 rad/s^2. It asks at what time will the object begin to slip.

Please help!

Homework Equations

Force of static friction = m*g*mu_static
Centripetal, or radial, acceleration = v^2/r
v = sqrt(r*g*mu_static)

The Attempt at a Solution

The idea is that up to a certain velocity, the static friction force should prevent the object from moving outside of the circular trajectory. So, I set v^2/r <= g*mu_static, and solved for v, which is about .82112 m/s. The reason why I went with radial acceleration was because I know that the overall centripetal acceleration will combat against friction's acceleration and force, and I know that the magnitude of that acceleration is sqrt(a_radial^2 + a_tangential^2). I'm having a hard time establishing relationships because it's hard for me to visualize what is affecting what.

 

[jbsteele]

For the second part, I'm not sure how to approach this. I know that a = alpha*t, but am not sure how to apply it here. I'm thinking that I need to integrate the acceleration with respect to time to find the velocity, and then plug in the velocity into the equation v^2/r <= g*mu_static, but I'm not sure if that's correct. Any help is really appreciated!