Ann, Sam and you take a trip to the orbiting International Space Station, bringing along a hula hoop that is 1.02 m in diameter (= 2R) and has a mass Mh = 2.0 kg. You are watching from a viewport while Sam and Ann go outside for a (space) walk. You see both Sam and Ann motionless in space a distance D = 23.6 m apart and oriented in the same direction as each other, parallel head to toe along a line joining their centers of mass. Ann is holding the hula hoop motionless, centered on her body, but decides to throw it ‘up’ to Sam, giving the hula hoop a spin in the process. You observe the hula hoop moving between Ann and Sam with a translational speed vh = 3.9 m/s and spin of frequency fh = 2.5 revolutions/sec. Sam catches the hula hoop and holds it as shown. Sam has a mass (including spacesuit) Ms = 90 kg and a moment of inertia about his axis Is = 0.65 kg-m2. Ann has a mass (also including space suit) 2
MA = 65 kg and a moment of inertia about her axis of IA = 0.55 kg-m .
After Sam catches the hula hoop, you see both Sam and Anne spinning and moving away from each other. Calculate their translational velocities vA and vS and the frequencies fS and fA at which they are spinning. Solve for each quantity first in terms of symbols; don’t plug in the numerical values you are provided until the last possible step.
To calculate the translational velocities vA and vS and the spinning frequencies fS and fA of Sam and Ann after Sam catches the hula hoop, we can apply the principle of conservation of angular momentum and conservation of linear momentum.
1. Conservation of angular momentum:
The total angular momentum of the system (Sam, Ann, and the hula hoop) remains constant before and after the throw. The initial angular momentum is given by:
L_initial = IA * ωA_initial + Is * ωS_initial,
where IA is the moment of inertia of Ann, ωA_initial is her initial angular velocity, Is is the moment of inertia of Sam, and ωS_initial is his initial angular velocity.
Since the hula hoop is initially at rest with Ann and is thrown to Sam without any external torques acting, the initial angular momentum of the hula hoop is zero.
After Sam catches the hula hoop, the final angular momentum is given by:
L_final = IA * ωA_final + Is * ωS_final,
where ωA_final and ωS_final are the final angular velocities of Ann and Sam, respectively.
Since the hula hoop is spinning when Sam catches it, its angular momentum is non-zero in the final state. We can relate the final angular velocities to the initial angular velocities using the conservation of angular momentum equation:
L_initial = L_final
IA * ωA_initial + Is * ωS_initial = 0 + (IA + Is) * ω_final,
where ω_final is the common final angular velocity of Sam and Ann.
2. Conservation of linear momentum:
The total linear momentum of the system remains constant before and after the throw. The initial linear momentum is given by:
P_initial = MA * vA_initial + Ms * vS_initial,
where MA and Ms are the masses of Ann and Sam, respectively, and vA_initial and vS_initial are their initial linear velocities.
After Sam catches the hula hoop, the final linear momentum is given by:
P_final = MA * vA_final + Ms * vS_final,
where vA_final and vS_final are the final linear velocities of Ann and Sam, respectively.
Since there are no external forces acting on the system in space, the conservation of linear momentum equation can be written as:
P_initial = P_final
MA * vA_initial + Ms * vS_initial = MA * vA_final + Ms * vS_final.
Now, let's solve the equations above to find vA, vS, fA, and fS in terms of the given quantities.
First, let's find the final angular velocity ω_final:
From the conservation of angular momentum equation:
IA * ωA_initial + Is * ωS_initial = (IA + Is) * ω_final,
ω_final = (IA * ωA_initial + Is * ωS_initial) / (IA + Is).
Next, let's find the final linear velocities vA_final and vS_final:
From the conservation of linear momentum equation:
MA * vA_initial + Ms * vS_initial = MA * vA_final + Ms * vS_final.
To solve for vA_final and vS_final, we need to know the masses MA and Ms, as well as the initial linear velocities vA_initial and vS_initial. Plug in these values into the equation above to find vA_final and vS_final.
Finally, let's find the spinning frequencies fA and fS:
Since f (frequency) is related to ω (angular velocity) by the equation f = ω / (2π), we can find fA and fS by substituting the values of ω_final into the equations:
fA = ωA_final / (2π) and fS = ωS_final / (2π).
By substituting the appropriate values into the equations derived above, you can calculate the translational velocities vA and vS and the spinning frequencies fA and fS for Ann and Sam after Sam catches the hula hoop.