Yesterday, I went bowling and while I was waiting for my turn, I decided to compute the total kinetic energy of a rolling ball. Treat the ball as a solid sphere with a diameter of 22.0 cm and a mass of 5.00 kg, the ball rolls down the lane without slipping at a speed of 15.0 km/h. What did I find for the total kinetic energy

Ktot = 60.8 J

Explanation:

The total kinetic energy of a rolling ball, is composed by a rotational kinetic energy (due to the rotation around a diameter), plus a traslational kinetic energy, due to the translation of the center of  mass.The rotational kinetic energy can be written as follows:

       

where I = moment of  Inertia of a solid sphere, and  ω², the square of  the angular speed.If the ball rolls without slipping, there exists a fixed  relationship between the angular speed and the speed of the center of mass, as follows:

       

The traslational part, can be expressed as follows:

       

For a solid sphere, like a bowling ball, the moment of inertia about an axis passing through a diameter can be written as follows:

       

Replacing (2) and  (4) in (1), we have:

       

The total kinetic energy is just the sum of (3) and (5):

       

The speed of the center of  mass  is given in km/h, so it is needed to convert it to m/s, as follows:

       

Replacing (7) and m=5.00 kg in (6), we get the total kinetic energy of the rolling ball, as follows:

       

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