Boy Loses Contact with Ice Mound: Approximate Solution

[kapopka88]

A boy is initially seated on the top of a hemispherical ice mound of radius R = 2.6 m. He begins to slide down the ice, with a negligible initial speed (Fig. 8-47). Approximate the ice as being frictionless. At what height does the boy lose contact with the ice?

 

[Avodyne]

Ah, a classic problem!

What are your ideas to solve it? Can you say what is special about the position or velocity or acceleration at the point that contact is lost?

 

[kapopka88]

i really have no idea where to start. the velocity at that point is very small and the acceleration would be positive, the position would be a height less than 2.6 but that's all i can figure out.

 

[Avodyne]

Will the velocity really be "very small"? What would "very small" mean?

Before he leaves the ice, how is his acceleration in the radial direction related to his speed? What forces produce this acceleration? (Try drawing a free-body diagram.)

Are all those forces still present at the moment he loses contact?

 

[Avodyne]

and the only force acting on the boy is gravity.

After he leaves the ice, yes, but before he leaves there is another force.

 

[kapopka88]

oh centripetal force is also acting on him before he looses contact

 

[kapopka88]

v=sqrt(2g(r-h))

 

[Avodyne]

oh centripetal force is also acting on him before he looses contact

Well, I would call it the normal force of the ice on the boy.

 

[Avodyne]

v=sqrt(2g(r-h))

Yes. (Can you explain why this equation is valid?)

So, you have one relation between v and h.

At the moment he loses contact, the normal force of the ice on the boy is zero. So, his centripetal acceleration must be produced by the force of ? And the component of that force in the radial direction is ? (Why do we want the component in the radial direction?)

 

[Avodyne]

That's a sloppy way of saying it. I would say the centripetal acceleration is produced by the force of gravity. It's best not to confuse cause (force) with effect (acceleration).

 

[kapopka88]

so, centripetal force =0, so gravity must produce his centripetal acceleration. how do we find the radial component with no angle?

 

[kapopka88]

i set the centripetal force equal to gravity to solve for v, then plugged v into my first equation to get height but that was not right.

 

[Avodyne]

so, centripetal force =0, so gravity must produce his centripetal acceleration. how do we find the radial component with no angle?

There is an angle; the boy has moved off the top, and now a line from him to the center of the sphere makes an angle theta with respect to vertical. How is this angle related to h and R?

 

[Avodyne]

i set the centripetal force equal to gravity to solve for v, then plugged v into my first equation to get height but that was not right.

Show your work!

 

[kapopka88]

sin(theta)=h/r

 

[kapopka88]

so, so far i have F=mv^2/R, v=sqrt(2g(r-h)), and sin(x)=h/r

 

[Avodyne]

What is F?

 

[kapopka88]

the magnitude of the centripetal force

 

[Avodyne]

And that should be equal to ?

 

[Avodyne]

No, sorry. I think you need to find someone locally who can help you with some basics. (A force is not going to be equal to a length ...)