Explain in terms of angular momentum and rotational inertia, exactly how one "pumps up" a swing in the sitting position.

The spin cycle in washing machine spins the clothes in the washing machine making them move circular. the continious circular motion by the spin cycle gives centripetal and pseudecentripetal ( centrifugal ) force. after the completion of wash, the centrifugal force generated makes the water in the clothes get extracted due to high force ( caused by acceleration due to the spin cycle ) and make clothes dry. so, it primarily acts as moisture absorbent.   

Using the parallel axis theorem

We note the angular speed and also the system energy. Using these information we calculate the moment of inertia of irregular shaped body.

Io=2E/w²

When we have moment of inertia of irregular shaped body, we can then use parallel axis theorem to solve the moment of inertia if the body about the given axis.

I=Io+md²

We know that the moment of inertia of any rotating body is directly proportional to the square of the distance from its rotational axis. In the case of a flywheel, the position of an individual particle from the rotational axis matters for determining its moment of inertia. So, in order to maximize the moment of inertia, while minimizing its weight, we have to make a flywheel with the maximum possible radius while having gaps in the body, i.e. having spoke-like structures to hold the wheel. A...

I, personally loved this problem. Though I am not that much interested in playing sports on my own, I love the beauty of physics behind each sport, and the way physics beautifies each sport.

Before approaching the question, let's know how diving is scored in Olympic games. The judges score looking at the starting position, approach, height, flight, and entry. The higher jump with a maximum number of somersaults and a perfect entry with a minimal splash is always considered the best one. 

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The angular speed of the student will increase. When the dumbbells are dropped on the floor, no external torque acts on the student and the turntable, and therefore the angular momentum must be conserved.
When the dumbbells are dropped, the rotational inertia of the student and turntable decreased. To compensate for the decrease in rotational inertia the angular speed of the student and the turntable must increase.

The initial angular momentum due to the spin in the football ensures that any angular momentum due to the action of external forces does not change the direction of the initial angular momentum significantly. This allows the ball to reach the catcher with the same orientation as it was when thrown by the passer.
In case of the arrow, the lack of initial spin allows external forces such as gravity to change the direction of the angular momentum vector. This is accomplished by moving the arrow...

Balance on a bicycle is a matter of constantly correcting against falls, and it's easier when the speed is higher because the inertia of moving forward overcomes the need for corrective actions.

In the air, there's nothing holding the body of the helicopter still - so the rotor blades spin one direction, but the torque applied to spin the rotors tries to spin the body of the helicopter in the opposite direction. The little rotor on the tail helps keep the body of the helicopter pointed in the same direction.