What is the translational speed of a solid ball of radius 0.5 m,that is initiall

Question

What is the translational speed of a solid ball of radius 0.5 m,that is initially at rest, after it rolls 5m down a hill that isinclined at 30 degrees to the horizontal. The moment ofinertia of a solid ball about an axis through its center of mass is2/5MR2 where M is its mass and R is itsradius. (Back of the book says the answer is 5.92 m/s, but Iwould like to know how they got this answer)

Also based on the above information find the rotational speed ofthe solid ball at the bottom of the hill. (Hint use V=r)

Explanation / Answer

this is a conservation of energy problem, the ball is atop a hillthat can be thought of as a triangle with an angle 300so the height is 5m(sin30) = 2.5 meters the initial gravitational potential energy (mgh) equals the finalkinetic energy due to translational and rotational motion. thus mgh = 1/2 I 2 + 1/2 m v2 mgh = 1/2 I 2 + 1/2 mr22 now we know everthing but which can be solved with algebraicmanipulation. 2 = mgh/(1/2 I + 1/2 m r2)

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