There are two spherical balls, each of radius 100 cm., lying on a perfectly horizontal floor and touching each other.
What is the diameter of the largest ball that can pass through the gap between the spheres and the floor ?

Considering an angle Theta with it's vertex in the center of any of the two larger spheres and it's aperture from the vertex to the floor and from the vertex to the center of the small sphere in the mentioned gap between spheres. By drawing an isoceles triangle from the three centers of each circle, and relating Theta to the new isoceles triangle, it gives out the relation 4*R ²=2*(R+X)²*(1-cos[180-2*(90-Theta)]), where X is the radius to be calculated. By makeing another triangle with sides R, R+X, √(R²+X²) and Theta between R and R+X, we get the relation Theta=ArcCos[((R+X)²-X²)/(2*R*(R+X))], so we finally get the transient equation: 1=((2*R²)/(R+X)²)+Cos[180-2*[90-ArcCos[((R+X)²-X²)/(2*R*(R+X))]]], solving this equation for X, it gives X=25cm, so the diameter of the sphere in the gap between spheres is 50cm

Posted by Antonio on 2003-11-22 23:31:49