91Ó°ÊÓ

A space telescope is shown in the figure. One of the reaction wheels of its attitude-control system is spinning as shown at \(10 \mathrm{rad} / \mathrm{s}\), and at this speed the friction in the wheel bearing causes an internal moment of \(10^{-6} \mathrm{N} \cdot \mathrm{m}\). Both the wheel speed and the friction moment may be considered constant over a time span of several hours. If the mass moment of inertia of the entire spacecraft about the \(x\) -axis is \(150\left(10^{3}\right) \mathrm{kg} \cdot \mathrm{m}^{2},\) determine how much time passes before the line of sight of the initially stationary spacecraft drifts by 1 arcsecond, which is \(1 / 3600\) degree. All other elements are fixed relative to the spacecraft, and no torquing of the reaction wheel shown is performed to correct the attitude drift. Neglect external torques.

Step by step solution

01

Understand the Problem

The problem involves determining how much time it takes for the spacecraft to drift by 1 arcsecond due to the frictional moment acting on a spinning reaction wheel. We need to consider the spacecraft's moment of inertia and the constant frictional torque applied.

02

Calculate the Angular Acceleration

Torque (\(\tau\)) is related to angular acceleration (\(\alpha\)) by:\[\tau = I \cdot \alpha\]where \(\tau = 10^{-6} \text{ N}\cdot\text{m}\) and \(I = 150 \times 10^{3} \text{ kg}\cdot\text{m}^2\). Solving for \(\alpha\) gives:\[\alpha = \frac{\tau}{I} = \frac{10^{-6}}{150 \times 10^{3}} = \frac{10^{-6}}{150000} \text{ rad/s}^2\]

03

Convert 1 Arcsecond to Radians

An arcsecond is equal to \(\frac{1}{3600}\) degree. Convert the angle from degrees to radians using:\[1\text{ degree} = \frac{\pi}{180}\text{ radians}\]Thus, an arcsecond in radians is:\[\text{arcsecond in radians} = \frac{1}{3600} \times \frac{\pi}{180} \approx 4.848 \times 10^{-6}\text{ radians}\]

04

Determine the Time to Reach the Drift

Using the equation for angular displacement from rest with constant acceleration:\[\theta = \frac{1}{2}\alpha t^2\]where \(\theta = 4.848 \times 10^{-6} \text{ radians}\) and \(\alpha = \frac{10^{-6}}{150000} \text{ rad/s}^2\). Solving for \(t\) gives:\[t^2 = \frac{2\theta}{\alpha} = \frac{2 \times 4.848 \times 10^{-6}}{\frac{10^{-6}}{150000}}\]\[t^2 = 1454.4\text{ s}^2\]\[t = \sqrt{1454.4} \approx 38.12 \text{ seconds}\]

05

Solution Verification

Verify the calculations to ensure that each step is correct and that unit conversions were done properly, ensuring a logical approach to reach the final time of approximately 38 seconds.

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Key Concepts

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

Torque

Torque is a fundamental concept when studying angular motion, especially in systems involving rotation. It refers to the rotational equivalent of linear force. In simpler terms, torque can be thought of as a twist that causes an object to rotate around an axis.

Torque is calculated as the product of force and the distance from the point of application to the axis of rotation, also known as the lever arm. It's generally denoted by the letter \( \tau \), and measured in Newton-meters (N·m). In angular systems, torque plays the same role as force does in linear systems; it propels or hinders the rotational movement.

In this exercise, the frictional torque in the spacecraft's reaction wheel acts as a small but constant force, impacting the wheel's rotation over time.

Moment of Inertia

The moment of inertia is essentially the rotational analog of mass in linear motion. It quantifies an object's resistance to changes in its rotational motion. The moment of inertia depends not only on the mass of the object but also on how that mass is distributed with respect to the axis of rotation.

Given in units of kg·m², it is denoted by \( I \) in calculations. For an object like the spacecraft in the exercise, having a large moment of inertia means it requires more torque to change its rotational speed. The given moment of inertia for the spacecraft is \( 150 \times 10^{3} \text{ kg} \cdot \text{m}^2 \), highlighting its substantial resistance to angular acceleration due to the distribution of its mass.

Angular Acceleration

Angular acceleration describes the rate of change of angular velocity over time. It is akin to what linear acceleration is, but in the context of rotational motion. Denoted by \( \alpha \), angular acceleration is directly influenced by torque and moment of inertia according to the formula:

• \( \tau = I \cdot \alpha \)

In our scenario, the friction from the reaction wheel provides a constant torque, resulting in a steady angular acceleration. Calculating \( \alpha \) in this exercise reveals how incremental rotational changes occur in the spacecraft over a specified period.

Friction Moment

Friction moment, a specific instance of torque, arises from frictional forces. These forces act in opposition to an object's movement, causing a reduction in speed or altering its course.

In rotating systems, frictional moments can be generated by bearing frictions, air resistance, or contact with another surface. In our exercise, the friction moment of \( 10^{-6} \text{ N} \cdot \text{m} \) represents the small, constant drag force affecting the spacecraft's rotation. This friction gradually shifts the orientation of the spacecraft as time progresses.

Attitude Control

Attitude control involves regulating the orientation of a spacecraft or any vehicle. This control is crucial for ensuring that a spacecraft maintains its intended path, points in the right direction for scientific observations, orients properly for communication, and effectively utilizes onboard instruments.

Systems like reaction wheels, thrusters, or magnetic torquers help adjust or maintain the spacecraft's attitude. In our context, a reaction wheel that experiences frictional effects causes an unintended drift. Managing such frictional influence is vital for maintaining the desired orientation and stability over time.

Spacecraft Dynamics

Spacecraft dynamics refers to the motion analysis of spacecraft under the influence of various internal and external forces and torques. Understanding these dynamics is key for navigation, attitude control, and successful mission execution.

Our exercise highlights a scenario where the internal dynamics, specifically the frictional forces in a reaction wheel, influence the spacecraft's motion over time. Even small forces like friction can accumulate into significant changes, requiring precise control and adjustment measures. This exercise exemplifies a practical concern in spacecraft dynamics, where the inertial properties and external influences must be carefully considered to predict and control motion accurately.