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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

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

Step by step solution

01

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 \( t=0 \).
02

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 \( \omega_2 \).
03

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, \( \omega_3(t) = \frac{\Gamma t}{\text{I}_3} + \omega_{30} \).
04

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 \( \mathbf{e}_1 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

A triangular prism (like a box of Toblerone) of mass \(M,\) whose two ends are equilateral triangles parallel to the \(x y\) plane with side \(2 a,\) is centered on the origin with its axis along the \(z\) axis. Find its moment of inertia for rotation about the \(z\) axis. Without doing any integrals write down and explain its two products of inertia for rotation about the \(z\) axis.

Consider an arbitrary rigid body with an axis of rotational symmetry, which we'll call \(\hat{\mathbf{z}}\). (a) Prove that the axis of symmetry is a principal axis. (b) Prove that any two directions \(\hat{\mathbf{x}}\) and \(\hat{\mathbf{y}}\) perpendicular to \(\hat{\mathbf{z}}\) and each other are also principal axes. (c) Prove that the principal moments corresponding to these two axes are equal: \(\lambda_{1}=\lambda_{2}\).

Consider a rigid plane body or "lamina," such as a flat piece of sheet metal, rotating about a point \(O\) in the body. If we choose axes so that the lamina lies in the \(x y\) plane, which elements of the inertia tensor \(\mathbf{I}\) are automatically zero? Prove that \(I_{z z}=I_{x x}+I_{y y}\).

Show that the inertia tensor is additive, in this sense: Suppose a body \(A\) is made up of two parts \(B\) and \(C\). (For instance, a hammer is made up of a wooden handle wedged into a metal head.) Then \(\mathbf{I}_{A}=\mathbf{I}_{B}+\mathbf{I}_{C} .\) Similarly, if \(A\) can be thought of as the result of removing \(C\) from \(B\) (as a hollow spherical shell is the result of removing a small sphere from inside a larger sphere), then \(\mathbf{I}_{A}=\mathbf{I}_{B}-\mathbf{I}_{C}\).

A rigid body consists of three masses fastened as follows: \(m\) at \((a, 0,0), 2 m\) at \((0, a, a),\) and \(3 m\) at \((0, a,-a) .\) (a) Find the inertia tensor \(\mathbf{I} .\) (b) Find the principal moments and a set of orthogonal principal axes.

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