A ball and a solid cylinder, made of the same material, of the same mass roll without sliding

| September 26, 2021 | education

A ball and a solid cylinder, made of the same material, of the same mass roll without sliding at the same speed, determine how many times the kinetic energy of the ball is less than the kinetic energy of a solid cylinder

Data: msh (solid ball mass) = mts (solid cylinder mass); Vsh (solid ball speed) = Vts (solid cylinder speed).

Kinetic energy of the specified solid ball: Eksh = msh * Vsh ^ 2/2 + Jsh * ω ^ 2/2 = (msh * Vsh ^ 2 + 2/5 * msh * R ^ 2 * (Vsh / R) ^ 2) / 2 = (msh * Vsh ^ 2 + 2/5 * msh * Vsh ^ 2) / 2 = 7/5 * msh * Vsh ^ 2/2 = 7/10 * msh * Vsh ^ 2.

The kinetic energy of the specified solid cylinder: Ekts = mts * Vts ^ 2/2 + Jts * ω ^ 2/2 = (mts * Vts ^ 2 + mts * R ^ 2/2 * (Vts / R) ^ 2) / 2 = (mts * Vts ^ 2 + mts * Vts ^ 2/2) / 2 = 3/2 * mts * Vts ^ 2/2 = 3/4 * msh * Vts ^ 2.

The ratio of kinetic energies: k = Ekts / Eksh = 3/4 * msh * Vts ^ 2 / 7/10 * msh * Vsh ^ 2 = 3/4 / 7/10 = 3 * 10 / (4 * 7) = 15 / 14 ≈ $ 1.07

Answer: The kinetic energy of the specified solid ball is 1.07 times less.



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