91Ó°ÊÓ

The moment of inertia is a fundamental concept in rotational dynamics, describing how the mass of an object is distributed in space. It reflects the resistance of an object to changes in its rotational motion. Think of it as the rotational equivalent of mass in linear motion.
For a rotating object, such as a reel, the moment of inertia depends on the shape and distribution of mass relative to the axis of rotation.
In this exercise, the reel's moment of inertia is given as \(0.44 \, \text{kg} \cdot \text{m}^{2}\). This value is crucial because it allows us to calculate other key factors like angular acceleration, using the rigid body equations in rotational dynamics.
It also shows how much torque is required to reach a particular angular acceleration. A larger moment of inertia implies a greater resistance to acceleration, making it harder to change its rotational speed.

Angular acceleration describes how quickly an object's rotational speed changes over time. In simple terms, it's the rate of change of angular velocity.
It's denoted by \( \alpha \) and measured in radians per second squared \(\text{rad/s}^2\).
In our scenario, angular acceleration can be found using the relationship between net torque \( \tau_{net} \) and moment of inertia \( I \):

• \( \tau_{net} = I \cdot \alpha \)

For the reel:

• \( \tau_{net} = 0.60 \, \text{N} \cdot \text{m} \)
• \( I = 0.44 \, \text{kg} \cdot \text{m}^2 \)

Using these values, we calculated \( \alpha \approx 1.364 \, \text{rad/s}^2 \).
Understanding this concept helps describe how the reel's speed will change as the hose unwinds.

Torque is a measure of the twisting force that causes an object to rotate. It's similar to force in linear motion but related to rotation. Torque depends on:

• The amount of force applied\( F \)
• The distance from the axis of rotation\( r \), which is also known as the lever arm.

The formula for torque \( \tau \) is:

• \( \tau = F \cdot r \)

In the exercise, the tension in the hose creates a torque \( \tau_{tension} = 4.00 \, \text{N} \cdot \text{m} \), while friction provides a resisting torque \( \tau_{friction} = 3.40 \, \text{N} \cdot \text{m} \).
The net torque \( \tau_{net} \) is the difference between these two, resulting in \( 0.60 \, \text{N} \cdot \text{m} \). This net torque is what causes the reel to rotate, unwinding the hose.

Angular displacement refers to the angle through which an object rotates. It's measured in radians and indicates how much rotation has occurred.
In this problem, we calculated the angular displacement \( \theta \) using the length of the unwound hose and the radius of the reel. The relationship is given by:

• \(\theta = \frac{L}{r}\)

Where \(L\) is the length of the hose and \(r\) is the radius.
For the given values:

• \(L = 15.0 \text{ m}\)
• \(r = 0.160 \text{ m}\)

This results in \( \theta = 93.75 \, \text{rad} \). Understanding angular displacement helps us track the rotational progress as the hose unwinds.