Finding Angle Which Cube Falls Off Sphere

[apierron]

A cube of polished steel sits on top of a highly polished and waxed hemisphere of polycarbonate (radius 92 cm). For these two surfaces interacting, the coefficient of kinetic friction is the same as the coefficient of static friction. The cube is given a very small 'nudge' perpendicular to the center of the cube and tangent to the top of the hemisphere. The cube starts to move with an initial speed. As the cube slides down the side of the sphere, it eventually loses contact with the surface of the sphere.



a.) Assume μ is zero and that the initial speed is also zero to solve for the angle (relative to the vertical) at which the cube falls off of the sphere.

b.) How does your answer to a change if the initial speed is not zero, but is 50cm/s? Discuss your results qualitatively (what changes and why) and describe them quantitatively (solve for the angle relative to the vertical at which the cube falls off).

c.) Assume that μ is 0.1 and that the initial speed is zero to solve for the angle.

d.) How does your answer to c change if the initial speed is not zero, but is 50cm/s? Discuss you results qualitatively (what changes and why) and describe them quantitatively (solve for the angle relative to the vertical at which the cube falls off).



I tried drawing a free body diagram and maybe looking at the forces that act on the cube but then I just got confused. Am I supposed to relate uniform circular motion into any of this? I attached a picture of the problem.

 

[apierron]

I forgot, he also gave us the length of one side of the cube as 3cm

 

[apierron]

Okay so I took another look at this problem and I'm pretty sure I went about it all wrong. My new approach would be to take the ∫F ds, where F would be the sum of my forces and ds would be the change in arc length. I started working this out and I just don't know what to set my limits as and what I would set this equal to, so that I could solve for θ. I thought maybe I could set it equal to zero since the only work done on the object is gravity but I'm not sure. Can anyone tell me if I'm headed in the right directions and give any hints to what I'm having trouble with?

 

[The Anonymous]

Hi Amber

I suggest using summation of forces for circular motion as well as conservation of energy.
Since energy conservation and summation of radial forces hold at any point for the cube-sphere system, that should get you there.

 

[apierron]

Hi Amber

I suggest using summation of forces for circular motion as well as conservation of energy.
Since energy conservation and summation of radial forces hold at any point for the cube-sphere system, that should get you there.

Okay thanks! I think that will be way easier than what I was trying to do.

 

[The Anonymous]

Once you include the effects of friction, then you'll need to evaluate the path integral for the friction force from theta=0 to theta=theta critical, but that'll end up being a lot like the example I did in class last week.

cheers

 

[The Anonymous]

You may want to reference this discussion...https://www.physicsforums.com/showthread.php?t=39043