A sphere of radius r is rolling without sliding. What is ratio of rotational kinetic energy and total kinetic energy associated with the sphere?

Total kinetic energy, in this case of a rolling sphere, consists of two parts: rotational kinetic energy and translational kinetic energy. Rotational kinetic energy is that caused by rotation of the sphere around its axis of rotation while the translational kinetic energy arises due to linear motion of the sphere. In rolling motion, these two forms of energy are coupled through the condition that the sphere rolls without sliding, implying there is no relative motion at the point of contact with the surface.

Rotational kinetic energy for a solid sphere is a particular fraction of total kinetic energy. For the given problem, the rotational kinetic energy: total kinetic energy ratio was 2/7. This suggests that the rotational part contributes less to the total energy as compared to the translational part. The remaining 5/7 is all taken up by the translation kinetic energy.

This concept is important in understanding how energy is distributed in rolling systems. It has practical applications in fields like mechanics, engineering, and physics education, helping to analyze the motion of rolling objects such as balls, wheels, and gears. The ratio emphasizes the importance of both rotational and translational dynamics in rolling motion.

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