Opening/Closing Arms on Rotating Stool Practice Problems

A diver spins in mid-air during a dive. The diver's body can be modeled as a solid cylinder which is ≈ 0.65 M including the diver's head, whereas his arms can be approximated and treated as thin rods equal to 0.20 M for both individual arms, noting that M is the total mass of the diver's body. Calculate how much faster the diver spins when he pulls in his arms towards his body. Assume that the length of each arm "r" is twice the value of the radius "R" of the cylindrical body.

A figure skater has her arms extended outwards at her sides, while spinning at a rate of 1.2 rev/s. If she pulls her arms close to her body, the rotation speed will increase to 1.8 rev/s.

(a) Why does the rotation speed increase?

(b) Express her final moment of inertia in terms of her initial moment of inertia.

A ballet dancer performs pirouettes, rotating at: 1.5, 2.5, and 3.5 revolutions about a vertical axis while she is airborne. The dancer leaps in an "open" position with an initial moment of inertia of 𝐼₀ and a rotational frequency of 𝑓₀ = 1.0 rev/s. When the dancer pulls in their limbs towards their body, the inertia is reduced to 𝐼 and spins at frequency of 𝑓 before extending their limbs outwards once again before landing back on the ground. What is the change in the angular momentum during the dancer's pirouette? Neglect air resistance.

Consider a ballet dancer executing pirouettes with rotations of: 1.5, 2.5, and 3.5 revolutions about a vertical axis while in the air. The dancer remains airborne for roughly 0.70 seconds. Initially, the dancer leaps into the air with arms and legs extended in an "open" position with an initial moment of inertia of I₀ and a rotational frequency 𝑓₀ = 1.2 rev/s, maintaining this for 0.10 s. Next, the dancer pulls in their limbs towards their body, which reduces their inertia to 𝐼 and achieves a rotational frequency of 𝑓 which is sustained for 0.50 s. Lastly, the dancer returns to the "open" position for 0.10 s until landing on the ground. Calculate the minimum rotational frequency 𝑓 during the middle phase of the jump to ensure that the dancer completes a single and a triple pirouette successfully.