A massive sphere of mass m and radius descends rolling without sliding, from rest, a rail that has a looped part with radius R, as indicated in the following figure (a) What minimum height H must the rail have in order for Can the massive sphere complete the loop without falling? (b) What minimum height h must the rail have for a block of mass m, starting from rest, to complete the loop without falling by sliding frictionlessly along the rail? (c) Explain the results obtained.

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Central Force

Central force is the force that acts along the center of mass of one body to the center of mass of another body. Its direction from or away from a point on a body to another is called the center of the force. Such forces include gravitational force between two massive objects or electrostatic forces between two charged particles. So from atoms to planets, the central forces play an important role in shaping our daily lives. One of the remarkable features of central forces is that they are conservative, i.e. the total energy of a central force system can always be expressed as a gradient of a potential energy function.