Docking bay: A space station consists of a large circular ring of mass m r and r

Question

Docking bay: A space station consists of a large circular ring of mass mr and radius R, connected by thin spokes (of negligible mass) to a tiny (negligible size) but dense (not negligible mass) central hub of mass mh. Initially, the station is at rest, neither translating nor rotating. An unpowered space pod of mass mp (much smaller than mr ) drifts through space with speed vp, approaching a space station on a path exactly tangential to the ring. As the pod touches the ring, the station's docking mechanism suddenly grabs it and holds it in place at that point on the ring.

a.) What is the station's moment of inertia about its hub before the pod docks?

b.) As the pod approaches the station, what is its angular momentum relative to the station's hub?

c.) After the pod docks, at what rate (in radians/sec) does the station rotate?

For the remainder of this problem, use "" as the station's angular velocity after docking no matter what answer you found for part (c).

d.) What apparent weight would a person in the pod feel because of this rotation?

e.) The station has rotational thrusters: jets located around the outside of the ring, pointed tangentially to control the rotation of the station. How much work must these jets do on the station in order to stop the rotation?

f.) If the station has four jets, located 90o apart around the ring and each capable of exerting a tangential force F on the station, what total torque can they apply?

g.) How much time will the jets require to stop the rotation?

Explanation / Answer

Ok I will take you step by step through what you need to do analytically. My equations aren't working correctly on this computer so I won't be able to show them. If you need me to explain further please let me know and I will... So we have a problem that will involve conservation of momentum, the caveat is that since the center of the station is not fixed energy will be conserved causing a translational velocity as well. You are not asked to determine the translational velocity, but just keep in mind that it will exist. (If you really wanted to determine this you would use conservation of energy. KE of pod = KE_rotation + KE_translation, you would find KE_rotation = .5*I*(w)^2 and that is found by using conservation of angualr momentum so your only unknown would be KE_Translatoin.) Sorry for the rambling, now to continue with your problem... a.) Moment of inertia of a ring with a thin wall is, I = M*R^2. The moment of inertia of a point with zero radius is zero. With this, the total moment of inertia of the Docking station is I = M*R^2. b.) Angualr momentum is defined as L =I*w, where I is moment of inertia and w is angular velocity; it is also defined as L = p x r, or the cross product of the linear momentum of an object with a vector pointing from the object to the pont in question. For the following, i^ will represent the x direction with, j^ the y direction and k^ the z direction. Using vector notation, L = (mass_pod * velocity_pod)i x [ Distance_from_center_of_station_in_x_direction i^ + Distance_from_center_of_station_in_x_direction j^] c.) This will involve conservation of angular momentum. Using the equation from step b, at the moment of impact L_pod = (mass_pod * velocity_pod)i^ x (Radius of ring)j^ just set resulting value equal to the angular momentum of the docking station just after the pod docks which is, L_after_docking = (mass_pod * R^2*w) + (mass_ring * R^2*w) and then just solve for w. You shoud get w = (velocity_pod * mass_pod) / (R*(mass_ring+mass_pod)) d.) The aparent weigh is jsut the force the astronaut feels while being in the pod, since the motion is angular he will be subject to centripital force which is equal to, m_astronaut*R_ring*w^2 e.) Work is equal to change in energy so we need to know how much is required to take the rotational velocity of the system to zero. This will be equal to: .5 * I_system * w^2 f.) Total torque would be 4*(F x r) or 4*F*R_ring g.) Since rotational work is equal to torque times theta, and we are assuming a constant force therefore constatn acceleration, we will use the rotationl motion equations. Specfically, theta = w*t + .5 * alpha * t^2 we know that alpha is equal to the torque divided by the moment of inertia so substitue that in, and torque was found in step f. We also know that work is equal to torque * theta so if you know how much work is required you can find theta then jsut sovle for t. so your equation will be: (Work / torque) = w * t +.5 * (torque / I) * t^2 solve for t. I hope this helps!

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