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Rotational Rate During a Somersault Dive
To find the rotational rate \( \omega \) during the somersault, we use the formula for rotational kinetic energy:
\[ K = \frac{1}{2} I \omega^2 \]Where:
- \( K \) is the rotational kinetic energy.
- \( I \) is the moment of inertia.
- \( \omega \) is the angular velocity.
Given:
- Rotational kinetic energy, \( K = 100 \) J
- Moment of inertia, \( I = 9.0 \) kg·m\(^2\)
We can rearrange the formula to solve for \( \omega \):
\[ \omega = \sqrt{\frac{2K}{I}} \]Substitute the given values:
\[ \omega = \sqrt{\frac{2 \times 100}{9.0}} \] \[ \omega \approx \sqrt{\frac{200}{9.0}} \] \[ \omega \approx \sqrt{22.22} \] \[ \omega \approx 4.71 \, \text{rad/s} \]So, the rotational rate during the somersault is approximately \( 4.71 \) rad/s.