To get a flat, uniform cylindrical satellite spinning at the correct rate, engineers fire four tange

To get a flat, uniform cylindrical satellite spinning at the correct rate, engineers fire four tangential rockets as shown in Fig. 10-56. If the satellite has a mass of 3600 kg, a radius of 4.0 rn, and the rockets each add a mass of 250 kg, what is the required steady force of each rocket if the satellite is to reach 32 rpm in 5.0 min, starting from rest? Cylindrical Satellite Calculation -------------------------------------------- Given: Mass of the satellite (m_satellite) = 3600 kg Radius of the satellite (r) = 4.0 m Additional mass from rockets (m_rockets) = 250 kg each Total mass added by four rockets (m_total_rockets) = 4 * 250 kg = 1000 kg Final angular velocity (ω) = 32 rpm Time to reach this velocity (t) = 5.0 min = 5 * 60 s = 300 s Calculating the final angular velocity in radians per second: 1. Convert rpm to radians per second: ω = 32 rpm * (2π rad / 1 min) * (1 min / 60 s) ω ≈ 32 * (2π / 60) rad/s ω ≈ 3.351 rad/s Calculating the moment of inertia (I) of the cylindrical satellite: 2. Moment of inertia for a solid cylinder: I = (1/2) * m * r² I = (1/2) * (3600 kg + 1000 kg) * (4.0 m)² I = (1/2) * (4600 kg) * (16 m²) I = 36800 kg·m² Calculating the angular acceleration (α): 3. Using the relation: α = (ω_final - ω_initial) / t Here, ω_initial = 0 (starting from rest) α = (3.351 rad/s - 0) / 300 s α ≈ 0.01117 rad/s² Calculating the torque (τ) required to achieve this angular acceleration: 4. Torque is related to moment of inertia and angular acceleration: τ = I * α τ = 36800 kg·m² * 0.01117 rad/s² τ ≈ 410.856 kg·m²/s² (N·m) Calculating the required force (F) of each rocket: 5. The torque provided by the force of one rocket is: τ = F * r Rearranging gives: F = τ / r F = 410.856 N·m / 4.0 m F ≈ 102.714 N Since there are four rockets: 6. Required force of each rocket (F_rocket): F_rocket ≈ 102.714 N Result: The required steady force of each rocket is approximately 102.71 N.