Sliding Off The Globe

So we can use energy to solve this as well as dynamics. We know that when the person is going to fall there is no normal force. And since it is a hemisphere the person also experiences a centripetal force.

$Fc = mv^2/R = mgcosϴ$

From here we can rearrange and find that the velocity at which the person falls off with is

$v^2 = gRcosϴ$

We will now use energy to manipulate things even further. We know the person has no initial velocity and that he is not moving. He/she initially has gravitational potential energy and then transforms some of that gravitational potential energy into kinetic energy. Using geometry we can find the height the person is at when he starts and when he slips.

$mgR = mgRcosϴ + mv^2/2$

we can substitute in the value of v^2 that we found earlier.

$mgR = mgRcosϴ + mgRcosϴ/2$

We can now factor out $mgR$ and we will get a new equation that is

$1 = cosϴ + cosϴ/2$

Using basic mathematics we can find the lowest common denominator and solve for $cosϴ$ to get

$cosϴ = 2/3$

And by doing the inverse cosine on both sides we get

$ϴ = cos^-1(2/3)$

$ϴ = 48$ degrees

The answer shown in this explanation is only approximate and not exact.