/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 111 The primary structure of a propo... [FREE SOLUTION] | 91影视

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Chapter 7: Problem 111

The primary structure of a proposed space station consists of five spherical shells connected by tubular spokes. The moment of inertia of the structure about its geometric axis \(A-A\) is twice as much as that about any axis through \(O\) normal to \(A-A\). The station is designed to rotate about its geometric axis at the constant rate of 3 rev/min. If the spin axis \(A-A\) precesses about the \(Z\) -axis of fixed orientation and makes a very small angle with it, calculate the rate \(\psi\) at which the station wobbles. The mass center \(O\) has negligible acceleration.

Short Answer

Expert verified
The precession rate \( \psi \) is approximately twice the small angle \( \theta \).

Step by step solution

01

Identify Known Quantities

The space station consists of spherical shells, and we know it rotates at 3 revolutions per minute (3 rev/min). We are asked to find the rate at which the station wobbles, denoted by \( \psi \). Also, the moment of inertia about axis \( A-A \) is twice that about any axis through \( O \) normal to \( A-A \).
02

Calculate the Moment of Inertia Relation

Let the moment of inertia about the geometric axis \( A-A \) be \( I_A \), and the moment of inertia about an axis through \( O \) normal to \( A-A \) be \( I_O \). According to the problem, \( I_A = 2 I_O \).
03

Use Gyroscopic Precession Formula

The formula for the gyroscopic precession rate \( \psi \) when the spin vector makes a small angle with the precession axis is given by \( \psi = \frac{L \sin \theta}{I_O \omega} \), where \( L \) is the angular momentum about \( A-A \), \( \theta \) is the small angle (very small, hence \( \sin \theta \approx \theta \)), and \( \omega \) is the angular velocity.
04

Calculate Angular Velocity

Convert the angular velocity from revolutions per minute to radians per second. Use the formula \( \omega = 3 \times \frac{2\pi}{60} \), yielding \( \omega = \frac{\pi}{10} \text{ rad/s} \).
05

Express Precession Rate in Terms of \( I_A \) and \( I_O \)

Using the relation \( L = I_A \omega \), we substitute in the gyroscopic precession formula: \( \psi = \frac{I_A \omega \theta}{I_O \omega} = \frac{2I_O \omega \theta}{I_O \omega} = 2 \theta \).
06

Conclude the Precession Rate

Since the angle \( \theta \) is very small, the precession rate \( \psi \) is approximately twice that of the small angle itself, assuming the angle \( \theta \) remains constant over time.

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Key Concepts

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

Moment of Inertia
The concept of moment of inertia is crucial in understanding how space stations or any rotating bodies behave. Moment of inertia, often denoted as \( I \), is a measure of an object's resistance to changes in its rotational motion. It depends on the mass distribution relative to the axis of rotation.
In the case of the space station, the moment of inertia about the geometric axis \( A-A \) is given as twice that about any axis through \( O \) normal to \( A-A \).
  • The formula for moment of inertia for a simple object can be \( I = mr^2 \), where \( m \) is the mass and \( r \) is the distance from the rotation axis.
  • In more complex systems like our space station, we calculate \( I \) by considering the distribution of mass across its entire structure.
This property indicates how hard it is to change the station's spinning speed, as well as how it behaves when subjected to external torques, such as those causing precession.
Angular Velocity
Angular velocity describes how fast an object rotates or revolves. For rotational systems like a space station, it's a pivotal factor.
In the exercise, the space station rotates about its geometric axis at a rate of 3 revolutions per minute, which converts into angular velocity in radians per second using \( \omega = 3 \times \frac{2\pi}{60} \).
  • One full revolution is \( 2\pi \) radians.
  • Converting to radians per second provides a consistent unit for physics calculations, which in this case results in \( \omega = \frac{\pi}{10} \) rad/s.
This angular velocity is vital in calculating other dynamics like angular momentum and gyroscopic precession.
Angular Momentum
Angular momentum is a measure of the extent an object will continue to rotate unless acted upon by an external torque. It is dependent on the object's moment of inertia and angular velocity, expressed as \( L = I_A \omega \).
For the space station, the angular momentum about the geometric axis \( A-A \) leverages both its moment of inertia \( I_A \) and angular velocity \( \omega \).
  • Angular momentum helps to understand the stability of the rotating system.
  • It provides critical insights into phenomena like gyroscopic precession.
This concept shows how the station maintains its rotational state and what occurs when it experiences precession, affecting its orientation in space.

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Most popular questions from this chapter

The earth-scanning satellite is in a circular orbit of period \(\tau .\) The angular velocity of the satellite about its \(y-\) or pitch-axis is \(\omega=2 \pi / \tau,\) and the angular rates about the \(x\) - and \(z\) -axes are zero. Thus, the \(x\) -axis of the satellite always points to the center of the earth. The satellite has a reaction-wheel attitude-control system consisting of the three wheels shown, each of which may be variably torqued by its individual motor. The angular rate \(\Omega_{z}\) of the \(z\) -wheel relative to the satellite is \(\Omega_{0}\) at time \(t=0,\) and the \(x\) - and \(y\) -wheels are at rest relative to the satellite at \(t=0 .\) Determine the axial torques \(M_{x}, M_{y},\) and \(M_{z}\) which must be exerted by the motors on the shafts of their respective wheels in order that the angular velocity \(\omega\) of the satellite will remain constant. The moment of inertia of each reaction wheel about its axis is \(I\). The \(x\) and \(z\) reaction-wheel speeds are harmonic functions of the time with a period equal to that of the orbit. Plot the variations of the torques and the relative wheel speeds \(\Omega_{x}, \Omega_{y},\) and \(\Omega_{z}\) as functions of the time during one orbit period. (Hint: The torque to accelerate the \(x\) -wheel equals the reaction of the gyroscopic moment on the \(z\) -wheel, and vice versa.)

The collar and clevis \(A\) are given a constant upward velocity of 8 in./sec for an interval of motion and cause the ball end of the bar to slide in the radial slot in the rotating disk. Determine the angular acceleration of the bar when the bar passes the position for which \(z=3\) in. The disk turns at the constant rate of 2 rad/sec.

Each of the identical wheels has a mass of \(4 \mathrm{kg}\) and a radius of gyration \(k_{z}=120 \mathrm{mm}\) and is mounted on a horizontal shaft \(A B\) secured to the vertical shaft at \(O .\) In case \((a),\) the horizontal shaft is fixed to a collar at \(O\) which is free to rotate about the vertical \(y\) -axis. In case \((b),\) the shaft is secured by a yoke hinged about the \(x\) -axis to the collar. If the wheel has a large angular velocity \(p=3600\) rev/min about its \(z\) -axis in the position shown, determine any precession which occurs and the bending moment \(M_{A}\) in the shaft at \(A\) for each case. Neglect the small mass of the shaft and fitting at \(O\)

The 5 -kg disk and hub \(A\) have a radius of gyration of \(85 \mathrm{mm}\) about the \(z_{0}\) -axis and spin at the rate \(p=1250\) rev/min. Simultaneously, the assembly rotates about the vertical \(z\) -axis at the rate \(\Omega=400\) rev/min. Calculate the gyroscopic moment \(\mathbf{M}\) exerted on the shaft at \(C\) by the disk and the bending moment \(M_{O}\) in the shaft at \(O .\) Neglect the mass of the shaft but otherwise account for all forces acting on it.

The rotor and shaft are mounted in a clevis which can rotate about the \(z\) -axis with an angular velocity \(\Omega .\) With \(\Omega=0\) and \(\theta\) constant, the rotor has an angular velocity \(\omega_{0}=-4 \mathbf{j}-3 \mathbf{k}\) rad/s. Find the velocity \(\mathbf{v}_{A}\) of point \(A\) on the rim if its position vector at this instant is \(\mathbf{r}=0.5 \mathbf{i}+1.2 \mathbf{j}+1.1 \mathbf{k} \mathrm{m} .\) What is the rim speed \(v_{B}\) of any point \(B ?\)
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