Question

An axially symmetric space station (principal axis \(\mathbf{e}_{3},\) and \(\lambda_{1}=\lambda_{2}\) ) is floating in free space. It has rockets mounted symmetrically on either side that are firing and exert a constant torque \(\Gamma\) about the symmetry axis. Solve Euler's equations exactly for \(\omega\) (relative to the body axis) and describe the motion. At \(t=0\) take \(\omega=\left(\omega_{10}, 0, \omega_{30}\right)\)

Short answer

\(\omega_1(t) = \omega_{10}\), \(\omega_2(t) = 0\), \(\omega_3(t) = \frac{\Gamma t}{\text{I}_3} + \omega_{30}\).

Step-by-step solution

Identify the Given Information

We have an axially symmetric space station with symmetrical axes and a constant torque around the symmetry axis. The Euler's equations for rotational motion under axial symmetry are given by:1. \( rac{d}{dt}( ext{I}_1 \omega_1) - ext{I}_2 \omega_2 \omega_3 = 0 \)2. \( rac{d}{dt}( ext{I}_2 \omega_2) - ext{I}_1 \omega_1 \omega_3 = 0 \)3. \( rac{d}{dt}( ext{I}_3 \omega_3) = \Gamma \)Where \( \text{I}_1 = \text{I}_2 \) because the body is symmetric, and \( \Gamma \) is the torque applied about the symmetry axis \( \mathbf{e}_3 \). Initial conditions are \( \omega = (\omega_{10}, 0, \omega_{30}) \) at \( t=0 \).

Solve for \(\omega_1(t)\) and \(\omega_2(t)\)

From the Euler equations: \[ \frac{d}{dt}( ext{I}_1 \omega_1) = \text{I}_2 \omega_2 \omega_3 \] Given \( \omega_2 \) is initially zero, and remains zero, as there is no initial angular velocity nor torque in this direction due to symmetry. Thus, \( \omega_1(t) = \omega_{10} \) remains constant because there's no component of torque or angular velocity creating a change in \( \omega_1 \) without \( \omega_2 \).

Solve for \(\omega_3(t)\) Using Torque

From the third Euler equation: \( \frac{d}{dt}( ext{I}_3 \omega_3) = \Gamma \)Integrating with respect to time gives:\( \text{I}_3 \omega_3(t) = \Gamma t + C \)To find \( C \), use the initial condition \( \omega_3(0) = \omega_{30} \):\( C = \text{I}_3 \omega_{30} \).Thus, \( \omega_3(t) = \frac{\Gamma t}{\text{I}_3} + \omega_{30} \).

Describe the Motion

The angular velocity \( \omega \) about the body axis changes over time: \( \omega_1 \) remains constant at \( \omega_{10} \), \( \omega_2 \) stays at 0, and \( \omega_3 \) increases linearly with time due to the constant torque exerted on the symmetric axis. Therefore, the space station spins faster around its symmetry axis \( \mathbf{e}_3 \) as time progresses, while maintaining a steady spin around \( \mathbf{e}_1 \).

Key concepts

Rotational Motion

Rotational motion refers to the movement of an object around a central axis. For the axially symmetric space station in our problem, it is influenced by the forces applied through the rockets, causing it to spin. In rotational motion, several key parameters come into play, such as angular velocity and torque. These help in understanding how the station moves and changes its position over time.
The principal axes are crucial in rotational motion as they define how the object spins. An object is in pure rotational motion if every particle of the object has a circular path around a fixed axis. The space station, being axially symmetric, maintains this circular uniformity, keeping certain rotational attributes constant while allowing others to change under external influence.
Euler's equations are particularly helpful in analyzing rotational motion. They describe how the axes' inertias (moments of inertia) relate to the angular velocities and any external torques applied. In a nutshell, rotational motion for the space station spins around these principles, maintaining equilibrium while reacting to applied forces.

Angular Velocity

Angular velocity, denoted as \omega\, is the rate at which an object rotates around an axis. In the context of the space station, the angular velocity determines how quickly it spins about its principal axes. It is expressed in terms of its components: \( \omega_1, \omega_2, \) and \( \omega_3 \) which correspond to rotations about each of the principal axes.
Understanding angular velocity is vital because it encapsulates the rotational state of the space station. For instance, in the given exercise, \( \omega_1 \) starts at a constant value \( \omega_{10} \), and \( \omega_3 \) shows linear growth over time due to the applied torque. These behaviors of \( \omega \) align with the rules set by Euler's equations, highlighting the dynamics of angular motion.
The constancy in \( \omega_1 \) indicates that there is no net external influence causing it to change, echoing the symmetrical nature of the station. Contrastively, \( \omega_3 \)'s linear increase is directly tied to the exerted torque \( \Gamma \) about the symmetry axis, illustrating how external forces manipulate angular velocity over time.

Axial Symmetry

Axial symmetry occurs when the physical properties of a structure, like the space station, are symmetrical about a particular axis. Here, the axis of symmetry is \( \mathbf{e}_3 \). This symmetry simplifies the analysis of rotational dynamics because it implies that the moments of inertia about two of the principal axes are equal (\( \lambda_1 = \lambda_2 \)).
Due to axial symmetry, the torque applied affects only the symmetry axis in this model, maintaining uniformity about the other axes. The symmetry causes \( \omega_2 \) to remain zero, emphasizing no change or rotation occurs about the secondary axis without initial conditions or external influence in that direction.
Axial symmetry streamlines the application of Euler's equations by reducing the complexity, allowing for straightforward solutions where certain components remain unchanged, while others alter predictably. Such symmetry hence plays a crucial role in simplifying and solving problems involving rotational dynamics, as it allows external forces to have a more predictable impact on the object.