# Bug lands on a frictionless sphere. Show that the Bug leaves head when

1. Oct 15, 2007

### gills

1. The problem statement, all variables and given/known data

A bug lands on top of the frictionless, spherical head of a bald man. It begins to slide down his head.

Show that the bug leaves the head when it has dropped a vertical distance 1/3 the radius of the head.

2. Relevant equations

Not sure. Maybe:

E_f - E_i = 0 since it is conserving energy (no friction)

and/or possibly

F=ma to calculate the normal force = 0 when the bug is 1/3 vertical h down the head somehow?

3. The attempt at a solution

Well, if i use the E_f - E_i = 0:

If the bug fell R/3 vertical displacement then its height at E_f = (2/3)R

[mg(2R/3) + (1/2)mv^2] - (mgR) = 0 ----->

not quite sure how to prove that the bug drops off the head via this method.

Tom

Last edited: Oct 15, 2007
2. Oct 15, 2007

### Staff: Mentor

One has to show that the bug leaves the surface at h = R/3.

At the point when and where the bug leaves the surface the normal (centrifugal) force would equal the component of gravity (centripetal force) pointing inward.

Construct a diagram which shows the bug starting at top of a sphere at R, and then it falls to h.

Determine the force balance equation and use the conservation of energy remembering the bugs starts with zero KE but with some GPE with respect to where it leaves the surface.

3. Oct 16, 2007

### gills

Would setting h = R/3 be incorrect because it mentions in the problem that it falls that vertical distance then falls, so it falls off the head at (2/3)R.

I'm confused on how to use the conservation laws with Normal force and the component gravity force of the bug. Any help would be great.

4. Oct 22, 2007

### gills

i could use a little help on this one too if anyone's around...

5. Oct 22, 2007

### BlackWyvern

6. Oct 22, 2007

### BlackWyvern

I don't understand this myself. I've looked at it for ages, and it just doesn't seem to make sense to me at all.

I would like some help as well, lol.

7. Oct 22, 2007

### gills

Yea, that link helped a little, but i'm still a little confused.

8. Oct 22, 2007

### Staff: Mentor

Sorry for the confusion. That is h measured from the top of the sphere rather than the horizontal diameter.

This should help

images by Gert Hamacher

One needs to find the relationship between the change in GPE (=mgh) and kinetic energy (using the tangential speed v), and using the expression for centripetal acceleration and geometry, find the relationship between h and R when the force on the sphere is zero, i.e. just before the mass leaves the surface.