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A ceiling fan has an angular acceleration of \(62 \mathrm{rad} / \mathrm{s}^{2}\) when acted on by a torque of \(8.3 \mathrm{~N} \cdot \mathrm{m}\). What is the moment of inertia of the fan?

Short Answer

Expert verified
The moment of inertia of the fan is approximately 0.134 kg·m².

Step by step solution

01

Identify Known Values

We know the angular acceleration \( \alpha \) is \( 62 \, \text{rad/s}^2 \) and the torque \( \tau = 8.3 \, \text{Nm} \).
02

Understand the Relationship

Torque \( \tau \) and angular acceleration \( \alpha \) are related by the equation \( \tau = I \cdot \alpha \), where \( I \) is the moment of inertia.
03

Rearrange the Formula

To find the moment of inertia \( I \), rearrange the formula: \( I = \frac{\tau}{\alpha} \).
04

Substitute the Values

Substitute the known values into the equation: \( I = \frac{8.3 \, \text{Nm}}{62 \, \text{rad/s}^2} \).
05

Calculate the Moment of Inertia

Perform the division: \( I = \frac{8.3}{62} \approx 0.134 \, \text{kg} \cdot \text{m}^2 \).

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Key Concepts

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

Angular Acceleration
Angular acceleration is a measure of how quickly an object changes its rotational speed. It is analogous to linear acceleration but instead applies to objects that rotate around an axis.

- **Units**: The standard unit for angular acceleration is radians per second squared ( \( \mathrm{rad/s}^{2} \)).
- **Direction**: Angular acceleration has both magnitude and direction. It can be positive or negative depending on whether the rotational speed is increasing or decreasing.The angular acceleration of a ceiling fan tells us how rapidly the speed of the fan's blades is changing. In our exercise, the ceiling fan experiences an angular acceleration of \( 62 \, \mathrm{rad/s}^{2} \), meaning it speeds up very quickly when the torque is applied.

Understanding angular acceleration is crucial for solving problems related to rotational motion, such as computing the moment of inertia, as we will see in the upcoming sections.
Torque
Torque can be thought of as the rotational equivalent of force. It measures how much a force acting on an object causes that object to rotate. Just like force causes linear acceleration, torque causes angular acceleration.

- **Formula**: The mathematical expression for torque is \( \tau = F \times r \), where \( F \) is the force applied, and \( r \) is the distance from the axis of rotation. In most cases, \( F \) needs to be perpendicular to \( r \) for maximum torque.
- **Units**: The unit of torque is the Newton meter (\( \mathrm{Nm} \)).In the ceiling fan example, a torque of \( 8.3 \mathrm{~Nm} \) causes the blades to start spinning faster. This torque originates from the motor of the fan, illustrating how the force applied to rotate an object translates into torque.

A robust understanding of torque helps in analyzing systems experiencing rotational dynamics, such as engines, see-saws, and indeed, ceiling fans.
Rotational Dynamics
Rotational dynamics is the branch of physics that deals with the motion of rotating objects. It describes how torques cause changes in rotational motion, just as forces cause changes in linear motion.

- **Key Equation**: An essential relationship in rotational dynamics is \( \tau = I \times \alpha \). Here, \( \tau \) is the torque, \( I \) is the moment of inertia, and \( \alpha \) is the angular acceleration.- **Moment of Inertia**: This is akin to mass in linear dynamics. The moment of inertia captures how mass is distributed relative to the axis of rotation and influences how torque affects rotational acceleration.
- In our problem: The ceiling fan, using the equation above, showed that with a torque of \( 8.3 \mathrm{~Nm} \) and angular acceleration of \( 62 \mathrm{~rad/s}^{2} \), the moment of inertia \( I \) was found to be approximately \( 0.134 \mathrm{~kg} \cdot \mathrm{m}^{2} \).
Understanding rotational dynamics allows us to apply these principles to various real-world problems, from machinery design to understanding the physics behind sports like gymnastics or discus throwing.

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Most popular questions from this chapter

Predict \& Explain Two children, Jason and Betsy, ride on the same merry-go- round. Jason is a distance \(R\) from the axis of rotation; Betsy is a distance \(2 R\) from the axis. (a) Is the angular speed of Jason greater than, less than, or equal to the angular speed of Betsy? (b) Choose the best explanation from among the following: A. The angular speed is greater for Jason because he moves more slowly than Betsy. B. The angular speed is greater for Betsy since she must go around a circle with a larger circumference. C. The angular speeds are the same because it takes the same amount of time for Jason and Betsy to complete a revolution.

To tighten a spark plug, it is recommended that a torque of \(15 \mathrm{~N} \cdot \mathrm{m}\) be applied. If a mechanic tightens the spark plug with a wrench that is \(25 \mathrm{~cm}\) long, what is the force necessary to create the desired torque?

How much time does it take for a spinning baseball with an angular speed of \(38 \mathrm{rad} / \mathrm{s}\) to rotate through \(15^{\circ}\) ?

Calculate A chef spins a disk of pizza dough over her head, giving it an angular speed of \(7.2 \mathrm{rad} / \mathrm{s}\). If the moment of inertia of the pizza dough is \(6.3 \times 10^{-6} \mathrm{~kg} \cdot \mathrm{m}^{2}\), what is its rotational kinetic energy? (Assume that the disk of dough is uniform.)

A rigid object rotates about a fixed axis. Do all points on the object have the same angular speed? Do all points on the object have the same linear speed? Explain.

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