A) \[2{{L}_{1}}\]
B) \[4{{L}_{1}}\]
C) \[6{{L}_{1}}\]
D) \[8{{L}_{1}}\]
Correct Answer: D
Solution :
[D]Although the problem asks us to compare angular momenta of the two dumbbells, \[L=I\omega ,\] and \[\omega \]is identical for both dumbbells. Therefore, this is really a problem asking us to compare the moments of inertia I |
\[{{I}_{1}}=\sum{{{r}^{2}}m}\] |
\[{{I}_{1}}={{\left( \frac{1}{2}d \right)}^{2}}m+{{\left( \frac{1}{2}d \right)}^{2}}m=\frac{{{d}^{2}}m}{2}\] |
Now let?s calculate \[{{I}_{2}}\]is 8 times \[{{I}_{1}}\] |
\[{{I}_{2}}=\sum{{{r}^{2}}m}\] |
\[{{I}_{2}}={{(d)}^{2}}2m+{{(d)}^{2}}2m\] |
\[{{I}_{2}}=2{{d}^{2}}2m=4{{d}^{2}}m\] |
\[{{I}_{2}}=8\left( \frac{{{d}^{2}}m}{2} \right)=8{{I}_{1}}\] |
Because the moment of inertia \[{{I}_{2}}\] is 8 times \[{{I}_{1}},\]\[{{L}_{2}}\]the angular momentum is\[8{{L}_{1}}\] as well. |
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