Opening/Closing Arms on Rotating Stool: Videos & 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 toward 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 1.5 and 3.5 pirouette successfully.

What is angular momentum, and how is it conserved?

A person sits on a rotating stool holding two 5 kg weights at a distance of 0.5 m from the axis. What is the moment of inertia of the system?

If a wheel is rotating at 300 RPM, what is its angular velocity in radians per second?

A skater spins with arms extended, then pulls them in. What happens to the moment of inertia and angular velocity?

Calculate the angular momentum of a rotating wheel with a moment of inertia of 10 kg·m² and an angular velocity of 2 rad/s.

Which sequence correctly outlines the steps to solve a problem involving angular momentum conservation on a stool?

How does the conservation of angular momentum explain the increased speed of a person spinning on a stool when they pull their arms in?

Why does a decrease in moment of inertia result in an increase in rotational speed?

A rotating system has a moment of inertia of 5 kg·m² and an angular velocity of 10 rad/s. If the moment of inertia changes to 2 kg·m², what is the new angular velocity?

How does the moment of inertia of a point mass change when its distance from the axis of rotation doubles?

What is the final angular velocity of a system if the initial moment of inertia is 8 kg·m², initial angular velocity is 3 rad/s, and final moment of inertia is 4 kg·m²?

Why is conservation of angular momentum crucial for a figure skater performing spins?

Which of the following statements about angular momentum is correct?

What is the moment of inertia of a 10 kg mass located 0.3 m from the axis of rotation?

Convert 600 RPM to angular velocity in radians per second.

A rotating disc changes its shape such that its moment of inertia decreases by half. What happens to its angular velocity?