A cylinder of radius 2.09 cm and a sphere of radius 6.47 cm are rolling without

Question

A cylinder of radius 2.09 cm and a sphere of radius 6.47 cm are rolling without slipping along the same floor. The two objects have the same mass. If they are to have the same total kinetic energy, what should the ratio of the cylinder's angular speed to the sphere's angular speed be?

Explanation / Answer

K.E. of cylinder = 0.5*m*v^2 + 0.5*I*w^2 => K(cylinder) = 0.5*m*v^2 + 0.5*(m*r^2/2)*(v/r)^2 => Kc = 0.5*m*v^2 + 0.5*m*v^2/2 => Kc = 0.75*m*v^2 K.E. of sphere = 0.5*m*v'^2 + 0.5*I*w'^2 => Ks = 0.5*m*v'^2 + 0.5*[(2/5)*m*r^2]*(v'/r)^2 => Ks = 0.5*m*v'^2 + 0.2*m*v'^2 => Ks = 0.7*m*v'^2 Ks = Kc => 0.75*v^2 = 0.7*v'^2 => v/v' = 0.966 w/w' = (v/v')*(r'/r) => w/w' = 0.966*(6.47/2.09) => angular speed of cylinder/angular speed of sphere = 2.99

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