Question

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

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

Step-by-step solution

Identify Known Values

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

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.

Rearrange the Formula

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

Substitute the Values

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

Calculate the Moment of Inertia

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

Key concepts

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.