Two astronauts, both 80 kg, are connected in space by a light cable. When they are 10 m apart, they spin about their center of mass with 6 rad/s. Calculate the new angular speed they'll have if they pull on the rope to reduce their distance to 5 m. You may treat them as point masses, and assume they continue to spin around their center of mass.

• A. ω_f = 1.5 rad/s
• B. ω_f = 4 rad/s
• C. ω_f = 6 rad/s
• D. ω_f = 24 rad/s

Answer: (D)

Verified step by step guidance

1. Identify the principle of conservation of angular momentum, which states that if no external torques act on a system, the initial angular momentum will be equal to the final angular momentum.
2. Calculate the initial angular momentum (L_i) using the formula L_i = I_i * ω_i, where I_i is the initial moment of inertia and ω_i is the initial angular speed. For point masses, I_i = m * r_i^2, where m is the mass of one astronaut and r_i is the initial distance from the center of mass (half of the total distance between them).
3. Determine the final moment of inertia (I_f) after the astronauts pull the rope to reduce their distance. Use the formula I_f = m * r_f^2, where r_f is the final distance from the center of mass (half of the new total distance).
4. Apply the conservation of angular momentum: L_i = L_f, which implies I_i * ω_i = I_f * ω_f. Solve for the final angular speed (ω_f) using the equation ω_f = (I_i * ω_i) / I_f.
5. Substitute the known values into the equation: m = 80 kg, r_i = 5 m, r_f = 2.5 m, ω_i = 6 rad/s, and solve for ω_f.