- ph ] 2 6 A ug 2 00 8 A sphere moving down the surface of a static sphere and a simple phase diagram

A sphere moving down the surface of a static sphere and a simple phase diagram. Abstract A small sphere placed on the top of a big static fric-tionless sphere, slips until it leaves the surface at an angle θ l = cos −1 2/3. On the other extreme, if the surface of the big sphere has coefficient of static friction , µ s → ∞, the small sphere starts rolling and continues to do so until it leaves the surface at an angle θ l = cos −1 10/17. In the case where, 0 ≤ µ s < ∞, we get a simple phase diagram. The three phases are pure rolling, rolling with slipping and detached state. One phase line separates pure rolling from rolling with slipping. This diagram is obtained when stopping angles for pure rolling are plotted against static friction coefficients µ s. Study in this article is restricted to the case when the mobile sphere starts at the top of the static sphere with infinitesimal kinetic energy. An interesting text book problem in the introductory mechanics([1], [2], [3]) is to determine the release point for a block of mass m. The block slides without friction on the surface of a static sphere of radius R. The block starts from " rest " , i.e. with in-finitesimal velocity, from the top of the sphere. The block looses contact with the sphere, by the time it travels a vertical distance R 3. This is equivalent to an angular displacement cos −1 2 3. This result draws attention. The angular displacement does not depend either on the mass of the block or, the radius of the sphere. The result is general. When there is friction on the surface of a sphere ([4], [5], [6]), the block leaving with infinitesimal velocity at the top of the static sphere, immediately stops. This dull situation changes to better as soon as the block is given a finite velocity at the top of the static sphere. The block may now continue to slip. It may also get released from the surface of the sphere below. Whether the block will leave the sphere below or, stick depends on the amount of friction. In that sense, the initial velocity and the static friction start playing with the block in fixing the fate of it. In an interesting report ([6]), the authors got an exact relation between …