Question

A space station is to provide artificial gravity to support long-term stay of astronauts and cosmonauts. It is designed as a large wheel, with all the compartments in the rim, which is to rotate at a speed that will provide an acceleration similar to that of terrestrial gravity for the astronauts (their feet will be on the inside of the outer wall of the space station and their heads will be pointing toward the hub). After the space station is assembled in orbit, its rotation will be started by the firing of a rocket motor fixed to the outer rim, which fires tangentially to the rim. The radius of the space station is \(R=50.0 \mathrm{~m},\) and the mass is \(M=2.40 \cdot 10^{5} \mathrm{~kg} .\) If the thrust of the rocket motor is \(F=1.40 \cdot 10^{2} \mathrm{~N},\) how long should the motor fire?

Short answer

Answer: The rocket motor should fire for approximately 63 minutes.

Step-by-step solution

Calculate the required rotational velocity

First, let's calculate the required rotational velocity (angular velocity) to provide the astronauts with an acceleration similar to that of Earth's gravity. The centripetal acceleration, \(a_c\), is given by \(a_c = \omega^2 R\), where \(\omega\) is the angular velocity and \(R\) is the radius. We are given that the centripetal acceleration must be equal to Earth's gravity, so \(a_c = 9.81 \: m/s^2\). Now we will solve for angular velocity: $$ 9.81 = \omega^2 (50.0) $$

Solve for angular velocity

Now solve for \(\omega\): $$ \omega = \sqrt{\frac{9.81}{50.0}} = 0.442 \: rad/s $$ The required rotational velocity (angular velocity) is found to be \(\omega = 0.442 \: rad/s\).

Calculate the angular momentum and torque

Next, we need to find the angular momentum (L) and the torque (τ) produced by the rocket motor. The angular momentum can be calculated using the moment of inertia (I) and angular velocity, such that \(L = I\omega\). The space station can be considered as a thin hoop with the moment of inertia given by \(I = MR^2\). $$ L = I\omega = (2.4\cdot 10^{5})(50.0)^2(0.442) = 2.64\cdot 10^7 \: kg\:m^2/s $$ The torque can be calculated using the thrust force and the radius: $$ \tau = FR = (1.40\cdot 10^{2})(50.0) = 7\cdot 10^3\: N\:m $$ We've found that the angular momentum of the space station is \(2.64\cdot 10^7 \: kg\:m^2/s\), and the torque provided by the rocket motor is \(7\cdot 10^3\: N\:m\).

Calculate the time required for the motor to fire

Finally, we can use the torque and the angular momentum to calculate the time required for the motor to fire to achieve the target angular velocity. The relationship between torque, time, and the change in angular momentum is given by \(\tau \Delta t = \Delta L\). $$ \Delta t = \frac{\Delta L}{\tau} = \frac{2.64\cdot 10^7}{7\cdot 10^{3}} = 3.77\cdot 10^3 \: s $$ The time required for the rocket motor to fire to achieve the desired angular velocity is 3.77\(\cdot 10^3\) seconds or approximately 63 minutes.

Key concepts

Centripetal Acceleration

When we talk about creating artificial gravity in space stations, centripetal acceleration plays a fundamental role. It's the inward-directed acceleration that keeps an object moving in a circular path.

To understand how this works within a space station, picture the station spinning like a carousel. As it rotates, everything in contact with the station moves along a curved path, requiring a force directed towards the center - this is provided by the structure of the station itself, which acts as our 'centripetal force', leading to what we perceive as gravity.

In the case of our space station exercise, the centripetal acceleration needed to mimic Earth's gravity (\(9.81 \text{ m/s}^2\)) dictates the rotational speed required. By setting the needed centripetal acceleration equal to Earth's gravitational acceleration and rearranging the formula for centripetal acceleration \(a_c = \frac{v^2}{R} = \frac{(R\text{\)\(\omega)^2}}{R}\) where \(v\) is linear speed and \(\omega\) is the angular velocity, we can solve for \(\omega\). This approach allows astronauts to 'stand' on the inside of the rim and feel as though they are experiencing normal gravity.

Angular Velocity

Angular velocity is a measure of the rate of rotation, specifying how fast an object spins around its axis. Think of the second hand on a clock—it has a constant angular velocity as it sweeps around the clock's face.

In relation to our exercise, the angular velocity is what we calculate to determine how fast the space station needs to spin to create the sensation of gravity. Using the centripetal acceleration formula, \(\omega^2 R = a_c\), we can figure out this necessary angular velocity. Once we have that \(\omega\), the station can be designed to spin at this specific rate.

Relating Angular Velocity to Real Life

If the space station spins too slowly, astronauts won't experience enough force to mimic gravity; spin it too fast, and they might feel too heavy or even be at risk due to the structural integrity of the station. Therefore, finding the right balance through calculation of angular velocity is critical for their wellbeing and the effectiveness of artificial gravity in space.

Rotational Dynamics

Rotational dynamics refers to the study of objects in rotation—including forces and torques that cause or change rotational motion. For our space station, understanding rotational dynamics is essential for calculating how long the rocket motor should fire to achieve the desired artificial gravity.

Applying Rotational Dynamics to the Space Station

Firstly, you need the moment of inertia, which depends on how the station's mass is distributed relative to its rotating axis. For a hoop or ring-like space station, it's \(I = MR^2\). The angular momentum (\(L\)) of the station is related to this inertia and the angular velocity, as \(L = I\omega\).

The torque (\(\tau\)) produced by the rocket motor is obtained from the force of thrust and the radius (\(\tau = FR\)). With these values, we determine the time taken to reach the required angular momentum using the motor's torque, thus understanding how to set the station spinning at the right speed to create artificial gravity. The relationship between torque, time, and angular momentum (\(\tau \Delta t = \Delta L\)) gives us the duration the rocket must fire to achieve the desirable living conditions for astronauts.

Applying rotational dynamics is crucial for ensuring that the motor fires just long enough to reach, but not exceed, the necessary speed for maintaining comfortable 'gravity' on a space station.