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How does the kinetic energy of a ballerina increase? Conservation of angular momentum: $$L_1=L_2 \implies I_1\omega_1=I_2\omega_2\quad\quad (1)$$ Pulling in your arms reduces moment of inertia $I$, since the same mass is now distributed over a volume closer to the spin centre, $I=\sum mr^2$. As you say, reducing $I$, so $I_2<I_1$, implies ...


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You were probably expected to note that the path of any point of the tire is a cycloid. At the point of contact with the ground, the tire is not moving. As it rises from the ground it is moving faster, with a speed that is $v(1+\sin \theta)$ where $v$ is the bike speed and $\theta$ is measured counterclockwise from horizontal. The centrifugal force is ...


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When it is half full, however, the water bottle rotates for one half-spin, and then it stops rotating. Why is this? This is why you don't want to ship oil across the ocean in a half-full oil tanker. If you do, you had better equip that tanker with some very good anti-slosh mechanisms. The same goes for trucks, trains, and spacecraft carrying fluid. ...


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EDIT: The "old" explanation below is quite informal, as gravity cannot produce a torque around the center of mass. See the new explanation below it. Humans lean forwards slightly as they run (more so if they run faster, and have to counter larger drag forces). This makes gravity exert a counter-torque that exactly balances the torque produced by action of ...


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I guess, The most important question is wheter you will accelerate the fluid when throwing the bottle. This and conservation of angular momentum will be much more important than dissipation of energy in the fluid. If it is full, the water, due to the constraints on the system, will be accelerated with the bottle when throwing. So it will just continue to ...



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