You sit on top of a perfect hemisphere of ice, balanced perfectly on the very top. A gust of wind blows you forward, and you slide without friction downwards along the curved surface. At what angle from the top of the hemisphere will you fly off the ice and fall through the air?
Hint: draw a big picture, and assign symbolic values like m, R, etc., to any quantities that you think might be relevant. Don't forget the force required to keep an object moving in a circular path ....