Question

It is well known that for a hollow, cylindrical shell rolling without slipping on a horizontal surface, half of the total kinetic energy is translational and half is rotational. What fraction of the total kinetic energy is rotational for the following objects rolling brainly

Answer 1

The fraction of the total kinetic energy that is rotational for a uniform sphere is 1/3.

What is kinetic energy?

• Kinetic energy is a form of energy that an object or a particle has by reason of its motion.
• If work, which transfers energy, is done on an object by applying a net force, the object speeds up and thereby gains kinetic energy.

Option (a) $f = \frac{0.5}{1.5} = \frac{1}{3}$

Option (b) uniform sphere

            $f = \frac{2}{7}$

C) uniform hollow sphere

           $f = \frac{2}{5}$

D) uniform hollow cylinder with inner radius R/2 and outer radius R

           $f = \frac{5}{13}$

that fraction of total energy as rotational energy is given as

     $f = \frac{\frac{1}{2}Iw^{2} }{\frac{1}{2}mv^{2} + \frac{1}{2} Iw^{2} }$

   $f = \frac{m k^{2}(\frac{v^{2} }{R^{2} }) }{mv^{2} + m k^{2}(\frac{v^{2} }{R^{2} } )}$

    $f = \frac{\frac{k^{2} }{R^{2} } }{1 + \frac{k^{2} }{R^{2} } }$

A) uniform Solid cylinder for cylinder we know that

         $\frac{k\\ ^{2} }{R^{2} }$  = 0.5

          $f = \frac{0.5}{1.5} = \frac{1}{3}$

B) uniform Sphere for sphere we know that

        $\frac{k^{2} }{R^{2} } = \frac{2}{5}$

    $f = \frac{0.4}{1.4} = \frac{2}{7}$

C) uniform hollow sphere for hollow sphere we know that

            $\frac{k^{2} }{R^{2} } = \frac{2}{3}$

         $f = \frac{\frac{2}{3} }{\frac{5}{3} } = \frac{2}{5}$

D) uniform hollow cylinder with inner radius R/2 and outer radius R for annular cylinder

             $\frac{k^{2} }{R^{2} } = \frac{5}{8}$

            $f = \frac{\frac{5}{18} }{\frac{13}{8} } = \frac{5}{13}$

Learn more about kinetic energy

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The complete question is -

It is well known that for a hollow, cylindrical shell rolling without slipping on a horizontal surface, half of the total kinetic energy is translational and half is rotational. What fraction of the total kinetic energy is rotational for the following objects rolling without slipping on a horizontal surface?Part AA uniform solid cylinder.Part BA uniform sphere.Part CA thin-walled hollow sphere.Part DA hollow, cylinder with outer radius R and inner radius R/2.