Respuesta :
Answer:
The total kinetic energy is 2.8m J. (NOTE: m is mass of the sphere)
Explanation:
The total kinetic energy of a sphere is given by the sum of the rotational kinetic energy and the translational kinetic energy. That is,
[tex]K_{Total} = K_{R} + K_{T}[/tex]
The rotational kinetic energy [tex]K_{R}[/tex] is given by
[tex]K_{R} = \frac{1}{2}I\omega^{2}[/tex]
Where [tex]I[/tex] is the moment of inertia
and [tex]\omega[/tex] is the angular velocity
The translational kinetic energy [tex]K_{T}[/tex] is given by
[tex]K_{T} = \frac{1}{2}mv^{2}[/tex]
Where [tex]m[/tex] is the mass
and [tex]v[/tex] is the translational speed (velocity)
∴ [tex]K_{Total} = \frac{1}{2}I\omega^{2} + \frac{1}{2}mv^{2}[/tex]
But, the moment of inertia [tex]I[/tex] of a sphere is given by
[tex]I = \frac{2}{5}mr^{2}[/tex]
Where [tex]m[/tex] is mass
and [tex]r[/tex] is radius
∴ [tex]K_{Total} = \frac{1}{2}\times \frac{2}{5}mr^{2} \omega^{2} + \frac{1}{2}mv^{2}[/tex]
[tex]K_{Total} = \frac{1}{5}mr^{2} \omega^{2} + \frac{1}{2}mv^{2}[/tex]
Also, [tex]\omega = \frac{v}{r}[/tex]
∴ [tex]\omega^{2} = \frac{v^{2} }{r^{2} }[/tex]
Then,
[tex]K_{Total} = \frac{1}{5}mr^{2} \times \frac{v^{2} }{r^{2} } + \frac{1}{2}mv^{2}[/tex]
[tex]K_{Total} = \frac{1}{5}mv^{2} + \frac{1}{2}mv^{2}[/tex]
∴ [tex]K_{Total} = \frac{7}{10}mv^{2}[/tex]
From the question, [tex]v = 2.00 m/s[/tex]
Then,
[tex]K_{Total} = \frac{7}{10}m(2.00)^{2}[/tex]
[tex]K_{Total} = \frac{7}{10}m\times 4.00[/tex]
[tex]K_{Total} = 2.8m J[/tex]
Hence, the total kinetic energy is 2.8m J. (NOTE: m is mass of the sphere)
