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 . (Intro1 figure) 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 $m/2$ 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 $I_{\rm cm} = mr^2$ , 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 .

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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 K_total of a dumbbell of mass when it is rotating with angular speed omega and its center of mass is moving translationally with speed . (Intro1 figure) Denote the dumbbell's moment of inertia about its center of mass by I_cm . Note that if you approximate the spheres as point masses of mass $m/2$ 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 $I_{\rm cm} = mr^2$ , 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, I_cm , and omega .

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