Question

A space station is to provide artificial gravity to support long-term habitation by 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 1.90 × 10^4 seconds.

Step-by-step solution

Determine the target angular velocity for artificial gravity

In order to provide artificial gravity, the centripetal acceleration at the outer wall of the station must be equal to the acceleration due to gravity on Earth (9.81 m/s²). The centripetal acceleration can be calculated using the formula: \(a_c = \omega^2R\), where \(a_c\) is the centripetal acceleration, \(\omega\) is the angular velocity, and \(R\) is the radius of the space station. Solving for \(\omega\), we get: \(\omega = \sqrt{\frac{a_c}{R}}\) Plugging in the values, \(a_c = 9.81 \mathrm{~m/s^2}\) and \(R = 50\mathrm{~m}\), we obtain: \(\omega = \sqrt{\frac{9.81\mathrm{~m/s^2}}{50\mathrm{~m}}} = 0.442\mathrm{~rad/s}\) So, our target angular velocity is 0.442 rad/s.

Calculate the moment of inertia of the space station

The space station is a large wheel, so we can assume it to be a circular disk. The moment of inertia (\(I\)) for a circular disk is calculated using the formula: \(I = \frac{1}{2}MR^2\), where \(M\) is the mass of the disk, and \(R\) is the radius. Given, \(M=2.40\times10^5\mathrm{~kg}\) and \(R=50.0\mathrm{~m}\), we can calculate the moment of inertia as follows: \(I = \frac{1}{2}(2.40\times10^5\mathrm{~kg})(50.0\mathrm{~m})^2 = 3.00\times10^8\mathrm{~kg\cdot m^2}\)

Calculate the torque exerted by the rocket motor

The torque (\(\tau\)) generated by the rocket motor can be calculated using the formula: \(\tau = F\times R\), where \(F\) is the force exerted by the rocket motor and \(R\) is the radius of the space station. Given, \(F=1.40\times10^2\mathrm{~N}\) and \(R=50.0\mathrm{~m}\), we can find the torque as: \(\tau = (1.40\times10^2\mathrm{~N})(50.0\mathrm{~m}) = 7.00\times10^3\mathrm{~N\cdot m}\)

Use Newton's second law for rotational motion to find the angular acceleration

Newton's second law for rotational motion states that the torque applied to an object is equal to its moment of inertia multiplied by its angular acceleration (\(\alpha\)). Mathematically, this can be written as \(\tau = I\alpha\). Solving for the angular acceleration, we get: \(\alpha = \frac{\tau}{I}\) Plugging in the values from Steps 2 and 3, we get: \(\alpha = \frac{7.00\times10^3\mathrm{~N\cdot m}}{3.00\times10^8\mathrm{~kg\cdot m^2}} = 2.33\times10^{-5}\mathrm{~rad/s^2}\)

Calculate the time needed to achieve the target angular velocity

The relationship between angular velocity, initial angular velocity (\(\omega_0\)), angular acceleration (\(\alpha\)), and time (\(t\)) is given by the formula: \(\omega = \omega_0 + \alpha t\) Initially, the space station is not rotating, so \(\omega_0 = 0\). We want to find the time (\(t\)) such that the angular velocity (\(\omega\)) reaches the target value of \(0.442\mathrm{~rad/s}\) determined in Step 1. So, we can rewrite the formula as: \(0.442\mathrm{~rad/s} = 0 + (2.33\times10^{-5}\mathrm{~rad/s^2})t\) Solving for \(t\), we get: \(t = \frac{0.442\mathrm{~rad/s}}{2.33\times10^{-5}\mathrm{~rad/s^2}} = 1.90\times10^4\mathrm{~s}\) The rocket motor should fire for approximately 1.90 × 10^4 seconds to achieve the target angular velocity and provide artificial gravity in the space station.

Key concepts

Centripetal Acceleration

Creating artificial gravity in a space station employs the principle of centripetal acceleration, which is the acceleration that acts on an object moving in a circular path, directed towards the center of the circle. Imagine you're on a carousel, the outward pull you feel is a result of centripetal force, required to keep you moving in a loop. For astronauts in a space station, this circular motion creates an inward pull that mimics gravity.
Using the formula for centripetal acceleration,
\( a_c = \frac{v^2}{R} \), or in terms of angular velocity,
\( a_c = \frac{v^2}{R} = \frac{(R\theta)^2}{R^2} = \frac{R^2\theta^2}{R^2} = \theta^2 R \),
we determine the necessary speed for the rotating station to generate a comfortable living environment for the crew. For the space station designed as a circular wheel with radius (
\( R = 50 \text{ m} \)), the required centripetal acceleration to simulate Earth's gravity (
\( a_c = 9.81 \text{ m/s}^2 \)) is achieved by setting the station in motion at an appropriate angular speed.

Moment of Inertia

The moment of inertia plays a crucial role in determining how much torque is needed to spin the space station. It represents the resistance to rotational motion, akin to how mass resists linear motion. For a spinning object, the mass distribution relative to its axis of rotation is key. The formula for the moment of inertia of a circular disk about its central axis is
\( I = \frac{1}{2}MR^2 \),
where
\( M \) is the mass of the disk and
\( R \) is the radius of the station. The higher the moment of inertia, the more torque is required to achieve the same angular acceleration. By calculating the space station's moment of inertia, engineers can determine the right amount of force needed to initiate and maintain its rotational motion.

Rotational Motion

Rotational motion is the movement of an object around a central point or axis. This is critical for generating artificial gravity in space stations through spinning. The entire space station must rotate to create a stable centrifugal force that pulls the astronauts against the outer edge, imitating the effect of gravity on their bodies. Establishing a smooth and consistent rotational motion involves considering the moment of inertia and the force needed to overcome it, and the resulting rotary effects on the residents of the station. Understanding these parameters ensures the station can be accelerated to, and maintained at, the desired angular velocity without structural or physiological complications.

Angular Velocity

Angular velocity is a measure of how quickly an object rotates or revolves about an axis; it's the rotational equivalent of linear velocity. For our space station, achieving a certain angular velocity is necessary to provide the 'feel' of gravity. By using the expression
\( \frac{9.81 \text{ m/s}^2}{50 \text{ m}} \), the desired angular velocity is obtained. Angular velocity helps us understand how different points on the station's circumference experience this artificial gravity, and it ensures that the living conditions within the rotating station remain comfortable and predictable for the astronauts.