Problem: This problem illustrates the two contributions to the kinetic energy of an extended object: rotational kinetic energy and translational kinetic energy. You are to find the total kinetic energy  of a dumbbell of mass when it is rotating with angular speed and its center of mass is moving translationally with speed .  Denote the dumbbell's moment of inertia about its center of mass by . Note that if you approximate the spheres as point masses of mass  each located a distance  from the center and ignore the moment of inertia of the connecting rod, then the moment of inertia of the dumbbell is given by , but this fact will not be necessary for this problem. Find the total kinetic energy of the dumbbell. Express your answer in terms of m, v, , and .

FREE Expert Solution

Translational kinetic energy:

$\boxed{\mathbf{K}{\mathbf{E}}_{\mathbf{T}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{m}{\mathbf{v}}^{\mathbf{2}}}$

Rotational kinetic energy:

$\boxed{{\mathbf{KE}}_{\mathbf{R}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}{\mathit{I}}_{\mathbf{c}\mathbf{m}}{\mathit{\omega }}^{\mathbf{2}}}$

Total kinetic energy:

$\boxed{\mathbf{KE}\mathbf{=}\mathit{K}{\mathit{E}}_{\mathbf{T}}\mathbf{+}\mathit{K}{\mathit{E}}_{\mathbf{R}}}$

The total kinetic energy: