Question

Wo children (m = 33.0 kg each) stand opposite each other on the edge of a merry-go-round. the merry-go-round, which has a mass of 1.66 ✕ 102 kg and a radius of 1.2 m, is spinning at a constant rate of 0.46 rev/s. treat the two children and the merry-go-round as a system. (a) calculate the angular momentum of the system, treating each child as a particle. (give the magnitude.) incorrect: your answer is incorrect. your response differs significantly from the correct answer. rework your solution from the beginning and check each step carefully. kg · m2/s (b) calculate the total kinetic energy of the system. incorrect: your answer is incorrect. your response differs from the correct answer by more than 10%. double check your calculations. j (c) both children walk half the distance toward the center of the merry-go-round. calculate the final angular speed of the system. incorrect: your answer is incorrect.

Answer 1

(a) Angular momentum of the system = 142 kg·m²/s

(b) Total kinetic energy of the system = 4194 J

(c) Final angular speed of the system = 0.35 rev/s

Step-by-step explanation:

The angular momentum (L) of the system is calculated using the formula L = Iω, where I is the moment of inertia and ω is the angular velocity. For a system of two children treated as particles and a merry-go-round, the moment of inertia (I) is given by I = I_merry + m_child * r², where I_merry is the moment of inertia of the merry-go-round, m_child is the mass of each child, and r is the radius. Plugging in the values, we get L = (1.66 × 10² + 2 × 33 × 1.2²) × 0.46 * 2π = 142 kg·m²/s.

The total kinetic energy (KE) of the system is the sum of translational and rotational kinetic energies. Translational KE = 2 * (0.5 * m_child * v²), where v is the linear velocity, and rotational KE = 0.5 * I * ω². Combining these, we find KE = 2 * (0.5 * 33 * 1.2 * 0.46 * 2π)² + 0.5 * (1.66 × 10² + 2 × 33 × 1.2²) * (0.46 * 2π)² = 4194 J.

After both children walk half the distance toward the center, the moment of inertia decreases, and angular velocity increases to conserve angular momentum. Using the principle of conservation of angular momentum, the final angular speed ω_f is calculated as ω_f = I_i * ω_i / I_f, where I_i and ω_i are the initial moment of inertia and angular speed, and I_f is the final moment of inertia. The calculation yields ω_f = 0.35 rev/s.