A space station has the form of a hoop of radius R = 15 m, with mass M = 6000 kg. Initially its center of mass is not moving, but it is spinning slowly with angular speed ù0 = 0.0018 radians/s.Then a small package of mass m = 5 kg is thrown by a spring-loaded gun toward a nearby spacecraft at an angle of è = 25° as shown; the package has a speed v = 40 m/s after launch.
Calculate the center-of-mass velocity of the space station and its rotational velocity after the launch.
angle of è = 25° as shown;
without the angle, impossible for me to decipher.
To calculate the center-of-mass velocity of the space station, we can use the conservation of momentum. The total initial momentum of the system is zero since the space station is not moving initially. After the launch, the package will have momentum in the x-direction and the space station will have momentum in the opposite direction to conserve momentum.
Step 1: Calculate the x-component of the package's momentum:
px = mvx = m * v * cos(θ)
= 5 kg * 40 m/s * cos(25°)
≈ 180.282 Ns
Step 2: Calculate the x-component of the space station's momentum:
-px = m_RV_R
= -6000 kg * V
where V is the center-of-mass velocity of the space station.
Step 3: Equate the x-component momenta and solve for V:
px = -px
180.282 Ns = -6000 kg * V
V = 180.282 Ns / -6000 kg
≈ -0.03 m/s
So, the center-of-mass velocity of the space station after the launch is approximately -0.03 m/s.
Now, to calculate the rotational velocity of the space station after the launch, we can use the conservation of angular momentum.
Step 4: Calculate the initial angular momentum of the system:
L_initial = I_initial * ω_initial,
where I_initial is the initial moment of inertia and ω_initial is the initial angular velocity.
Given that the space station has the form of a hoop, the moment of inertia of a hoop about its axis is I_hoop = m_R^2 (where m is the mass and R is the radius).
Thus, I_initial = I_hoop = m_R^2 = 6000 kg * (15 m)^2 = 1,350,000 kgm^2.
Since the space station is initially stationary, ω_initial = ω_0 = 0.0018 rad/s.
L_initial = 1,350,000 kgm^2 * 0.0018 rad/s = 2430 kgm^2/s
Step 5: Calculate the final angular momentum of the system:
L_final = I_final * ω_final,
where I_final is the final moment of inertia and ω_final is the final angular velocity.
After the launch, the package contributes to the system's total moment of inertia.
I_final = I_hoop + m * d^2,
where d is the distance between the center of mass of the hoop and the package. Since the hoop is a uniform hoop, d = R.
I_final = (6000 kg * 15 m^2) + (5 kg * 15 m^2)
= 1,350,000 kgm^2 + 1125 kgm^2
= 1,351,125 kgm^2
Step 6: Equate the initial and final angular momenta and solve for ω_final:
L_initial = L_final
2430 kgm^2/s = (1,351,125 kgm^2) * ω_final
ω_final = 2430 kgm^2/s / 1,351,125 kgm^2
≈ 0.0018 rad/s
So, the rotational velocity of the space station after the launch is approximately 0.0018 rad/s.
To calculate the center-of-mass velocity of the space station, we need to use the principle of conservation of momentum.
The initial momentum of the space station is zero, since its center of mass is not moving. After the package is thrown, the momentum of the system remains conserved.
The momentum of the package can be calculated as the product of its mass and velocity:
P_package = m * v
The momentum of the space station can be given by:
P_space station = M * V
where V is the velocity we need to calculate.
Since the momentum is conserved, we have:
P_package + P_space station = 0
Substituting the values, we get:
m * v + M * V = 0
Solving for V:
V = - (m * v) / M
Now, let's calculate the rotational velocity of the space station after the launch.
The rotational momentum can be calculated using the formula:
L = I * ω
where L is the rotational momentum of the space station, I is the moment of inertia, and ω is the angular velocity.
The moment of inertia of a hoop can be given by:
I = M * R^2
Substituting the values, we get:
I = M * R^2 = 6000 kg * (15 m)^2
Now, we can calculate the rotational momentum:
L = I * ω = (6000 kg * (15 m)^2) * 0.0018 radians/s
So, calculating V and L, we can find the center-of-mass velocity and the rotational velocity of the space station after the launch.