At What Point Does the Boy Lose Contact with the Ice?

[NilliNilli]

Homework Statement

A boy is initially seated on the top of a hemispherical ice mound of radius r=13.8m. He begins to slide down the ice, with an initial speed v_i=3m/sec. Approximate the ice as being frictionless. At what height does the boy lose contact with the ice?

Homework Equations

The Attempt at a Solution

My instructor gave the hint that it has to do with how force is relevant to centripetal force.
I first attempted finding the boy's direction and speed right as he started sliding off the ice by doing vector addition with the initial velocity (which I interpret as his speed in the x direction) and the centripetal acceleration ( interpreted as his speed in the y direction).
I found the centripetal acceleration to be v²/r = (3)²/13.8 = 0.652m/s². Then I found the initial angle θ=12.26°.

I kind of abandoned that since I didn't think I was approaching it the right way at all. Going off of my instructor's hint, I figured the boy has to start off with a centripetal force, since his acceleration is centripetal to start with, then since he's not attached to the sphere his acceleration become free-fall. I'm not sure how I should go about calculating at what point his acceleration stops begin centripetal. I'm thinking I have to determine when that happens then use s=rθ to find the arc length, since I'm not sure how else to determine distance on the sphere.

 

[PhizKid]

I would use energy concepts here. But in terms of forces (Newton's second law) how do you know the boy loses contact with the ice? Try expressing that critical condition first

 

[NilliNilli]

That's what I need help with. I need to express that in numbers and equations and I don't know how to do that for this problem.

 

[PhizKid]

Well, typically when an object loses contact with any surface, the Normal force becomes less than or equal to 0. So draw a force diagram and label all the forces on the boy first

 

[NilliNilli]

That was the first thing I did, actually.

 

[PhizKid]

So what equation did you get using Newton's second law and the forces you found?

 

[NilliNilli]

F=mg+n
I thought if the force was centripetal it could be
F=mg+n=mv²/r
I understand what you mean that when n=0, that will be when the boy lost contact with the ice, but I'm still not sure how to connect that to an equation where I can find a time or height that it happens.

 

[PhizKid]

Are you sure you have your signs correct? Where is the Normal force pointing and where is the force of gravity pointing?

 

[NilliNilli]

I have normal pointing up and gravity pointing down, assuming this is at the top of the sphere. Assuming this is right where the boy's about to lose contact, gravity would still be down and normal force would be pointing perpendicular to the sphere. The normal force's direction will consistently change until it's 0.

 

[haruspex]

Suppose he loses contact when he has traveled through an arc of angle theta. Through what height has he descended? What is his speed now? What centripetal force is needed there to just maintain contact? What forces are available to provide it?