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183_notes:examples:a_rod_rotating_not_around_its_center [2014/11/05 22:03] – pwirving | 183_notes:examples:a_rod_rotating_not_around_its_center [2014/11/06 02:32] (current) – pwirving | ||
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$K_{tot} = K_{trans} + K_{rot} = \frac{1}{2}(Mr^{2}_{CM} + I_{CM})\omega^{2}$ | $K_{tot} = K_{trans} + K_{rot} = \frac{1}{2}(Mr^{2}_{CM} + I_{CM})\omega^{2}$ | ||
- | $K_{trans} = \frac{1}{2}(Mr^{2}_{CM}$ | + | $K_{trans} = \frac{1}{2}Mr^{2}_{CM}$ |
- | $K_{rot} = \frac{1}{2}I_{CM})\omega^{2}$ | + | $K_{rot} = \frac{1}{2}I_{CM}\omega^{2}$ |
+ | |||
+ | $I = (\frac{1}{12})ML^{2}$ for a thin rod | ||
=== Solution === | === Solution === | ||
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$K_{tot} = K_{trans} + K_{rot}$ | $K_{tot} = K_{trans} + K_{rot}$ | ||
- | + | Substitute in equations for rotational kinetic energy and translational kinetic energy | |
$K_{tot} = \frac{1}{2}(Mr^{2}_{CM} + I_{CM})\omega^{2}$ | $K_{tot} = \frac{1}{2}(Mr^{2}_{CM} + I_{CM})\omega^{2}$ | ||
+ | Substitute in $(\frac{1}{12})ML^{2}$ for L as we are dealing with the inertia for a thin rod. | ||
$K_{tot} = \frac{1}{2}(Mr^{2}_{CM}\; | $K_{tot} = \frac{1}{2}(Mr^{2}_{CM}\; | ||
+ | |||
+ | Gather the M's out of both equations so that your equation now looks like: | ||
$K_{tot} = \frac{1}{2}M(r^{2}_{CM}\; | $K_{tot} = \frac{1}{2}M(r^{2}_{CM}\; | ||
+ | |||
+ | Insert values for the corresponding variables. | ||
$K_{tot} = \frac{1}{2}(.140\; | $K_{tot} = \frac{1}{2}(.140\; | ||
+ | |||
+ | Solve for $K_{tot}$ | ||
$K_{tot} = 1.75J$ | $K_{tot} = 1.75J$ | ||