Intro to Rotational Kinetic Energy Video Lessons

Concept

# 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 Ktotal of a dumbbell of mass m when it is rotating with angular speed v (Figure 1) Denote the dumbbell’s moment of inertia about its center of mass by Icm. Note that if you approximate the spheres as point masses of mass m/2 each located a distance r from the center and ignore the moment of inertia of the connecting rod, then the moment of inertia of the dumbbell is given by Icm = mr2 but this fact will not be necessary for this problemFind the total kinetic energy Ktot of the dumbbell.Express your answer in terms of m, v, Icm and w.

###### FREE Expert Solution

Total kinetic energy:

$\overline{){{\mathbf{K}}}_{\mathbf{t}\mathbf{o}\mathbf{t}}{\mathbf{=}}{{\mathbf{K}}}_{\mathbf{t}\mathbf{r}\mathbf{a}\mathbf{n}\mathbf{s}\mathbf{l}\mathbf{a}\mathbf{t}\mathbf{i}\mathbf{o}\mathbf{n}\mathbf{a}\mathbf{l}}{\mathbf{+}}{{\mathbf{K}}}_{\mathbf{r}\mathbf{o}\mathbf{t}\mathbf{a}\mathbf{t}\mathbf{i}\mathbf{o}\mathbf{n}\mathbf{a}\mathbf{l}}}$

Translational kinetic energy:

$\overline{){{\mathbf{K}}}_{\mathbf{t}\mathbf{r}\mathbf{a}\mathbf{n}\mathbf{s}\mathbf{l}\mathbf{a}\mathbf{t}\mathbf{i}\mathbf{o}\mathbf{n}\mathbf{a}\mathbf{l}}{\mathbf{=}}\frac{\mathbf{1}}{\mathbf{2}}{{\mathbf{m}}}_{\mathbf{t}\mathbf{o}\mathbf{t}\mathbf{a}\mathbf{l}}{{{\mathbf{v}}}_{\mathbf{c}\mathbf{m}}}^{{\mathbf{2}}}}$

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###### Problem Details

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 Ktotal of a dumbbell of mass m when it is rotating with angular speed v (Figure 1) Denote the dumbbell’s moment of inertia about its center of mass by Icm. Note that if you approximate the spheres as point masses of mass m/2 each located a distance r from the center and ignore the moment of inertia of the connecting rod, then the moment of inertia of the dumbbell is given by Icm = mr2 but this fact will not be necessary for this problem

Find the total kinetic energy Ktot of the dumbbell.