Question

An athlete with a mass of 54.0 kg is rotating head-over-heels in mid-air, at an angular speed of 3.00 revolutions per second, in a tucked position (that is, with arms, legs, and head pulled into a tight ball). In this position, the athlete can be modeled as a solid sphere with a radius of 0.600 m, rotating about its center. Just before landing, while still in the air, the athlete fully extends her arms and legs. In this position, she can be modeled as a long, thin rod, with a length of 2.00 m, the distance from her feet to her fingertips. The rod rotates about the center, with the axis of rotation perpendicular to the length of the rod.

a. Determine the athlete's moment of inertia in the tucked position.
b. Calculate the athlete's angular momentum in the tucked position.
c. Find the athlete's moment of inertia in the fully extended position.
d. Determine the athlete's angular momentum in the fully extended position.

Answer 1

The moment of inertia in the tucked position is 19.44 kg * m^2 and the angular momentum is 366.51 kg * m^2/s. The moment of inertia in the fully extended position is 26.67 kg * m^2 and the angular momentum is 502.65 kg * m^2/s.

Step-by-step explanation:

In the tucked position, the athlete can be modeled as a solid sphere rotating about its center. The moment of inertia of a solid sphere is given by the formula I = (2/5) * m * r^2, where m is the mass and r is the radius of the sphere. In this case, the moment of inertia can be calculated as: I = (2/5) * 54.0 kg * (0.600 m)^2 = 19.44 kg * m^2.

The angular momentum in the tucked position is given by the formula L = I * ω, where ω is the angular speed in radians per second. In this case, the angular momentum can be calculated as: L = 19.44 kg * m^2 * (3.00 rev/s * 2π rad/rev) = 366.51 kg * m^2/s.

In the fully extended position, the athlete can be modeled as a long, thin rod rotating about its center. The moment of inertia of a rod is given by the formula I = (1/3) * m * L^2, where L is the length of the rod. In this case, the moment of inertia can be calculated as: I = (1/3) * (2.00 m * 4.0 kg) * (2.00 m)^2 = 26.67 kg * m^2.

The angular momentum in the fully extended position can be calculated using the same formula as before: L = I * ω. In this case, the angular momentum can be calculated as: L = 26.67 kg * m^2 * (3.00 rev/s * 2π rad/rev) = 502.65 kg * m^2/s.