A merry go round is rotating at a constant angular velocity of 5.4 rad/s. What is the frequency of the merry go round in revolutions per minute?
Answers
Answer:
The angular momentum of a system of particles around a point in a fixed inertial reference frame is conserved if there is no net external torque around that point:
\[\frac{d\overset{\to }{L}}{dt}=0\]
or
\[\overset{\to }{L}={\overset{\to }{l}}_{1}+{\overset{\to }{l}}_{2}\,\text{+}\,\text{⋯}\,\text{+}\,{\overset{\to }{l}}_{N}=\text{constant}\text{.}\]
Note that the total angular momentum
\[\overset{\to }{L}\]
is conserved. Any of the individual angular momenta can change as long as their sum remains constant. This law is analogous to linear momentum being conserved when the external force on a system is zero.
As an example of conservation of angular momentum, (Figure) shows an ice skater executing a spin. The net torque on her is very close to zero because there is relatively little friction between her skates and the ice. Also, the friction is exerted very close to the pivot point. Both
\[|\overset{\to }{F}|\,\text{and}\,|\overset{\to }{r}|\]
are small, so
\[|\overset{\to }{\tau }|\]
is negligible. Consequently, she can spin for quite some time. She can also increase her rate of spin by pulling her arms and legs in. Why does pulling her arms and legs in increase her rate of spin? The answer is that her angular momentum is constant, so that
\[{L}^{\prime }=L\]
or
\[{I}^{\prime }{\omega }^{\prime }=I\omega ,\]
where the primed quantities refer to conditions after she has pulled in her arms and reduced her moment of inertia. Because
\[{I}^{\prime }\]
is smaller, the angular velocity
\[{\omega }^{\prime }\]
must increase to keep the angular momentum constant.
Two illustrations of a spinning ice skater. In figure a, on the left, the skater has her arms and one foot extended away from her body. She is spinning with angular velocity omega and L equals I times omega. In figure b, on the right, the skater has her arms and foot pulled close to her body. She is spinning faster, with angular velocity omega prime and L equals I prime times omega prime.
Answer:
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Explanation:
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