91Ó°ÊÓ

The moment of inertia, often denoted by the symbol \( I \), is a measure of an object's resistance to changes in its rotational motion. Just like mass in linear motion, moment of inertia is crucial in angular motion. It depends on both the mass of the object and the distribution of that mass relative to the axis of rotation.
The formula is generally expressed as \( I = \sum m_i r_i^2 \), where \( m_i \) is the mass of each particle in the object, and \( r_i \) is the distance from the axis of rotation to each particle.
This quantity is key in determining how easily an object, like our reel, can be started or stopped from spinning. In this hose reel problem, having a moment of inertia of \( 0.44 \, \text{kg} \cdot \text{m}^2 \) indicates how much torque is needed to change its angular velocity. A larger moment of inertia would mean a greater resistance to change.

Torque is the rotational equivalent of force and it determines how much an object will rotate with a certain force applied. Measured in Newton-meters (Nm), it's calculated as \( \tau = F \cdot r \cdot \sin(\theta) \), where \( F \) is the force applied, \( r \) is the distance from the pivot point, and \( \theta \) is the angle between the force and the arm.
In our exercise, the torque due to tension was calculated by \( \tau_T = T \times R \), where \( T \) (tension) was \( 25.0 \, \text{N} \) and \( R \) (radius) was \( 0.160 \, \text{m} \). This gives a torque of \( 4.0 \, \text{N} \cdot \text{m} \).
Friction exerts an opposing torque of \( 3.40 \, \text{N} \cdot \text{m} \), resulting in a net torque of \( 0.60 \, \text{N} \cdot \text{m} \). This net torque is essential to understand how the reel's rotation accelerates.

Angular acceleration refers to how quickly the rotation speed of an object changes. It's analogous to linear acceleration but in a rotational context, measured in \( \text{rad/s}^2 \).
From the equation \( \tau = I \alpha \), where \( \tau \) is net torque, \( I \) is moment of inertia, and \( \alpha \) is angular acceleration, we can solve for \( \alpha \) to determine how fast the reel starts turning. With a net torque of \( 0.60 \, \text{N} \cdot \text{m} \) and a moment of inertia of \( 0.44 \, \text{kg} \cdot \text{m}^2 \), the reel has an angular acceleration of \( 1.36 \, \text{rad/s}^2 \).
This indicates the reel is speeding up due to the applied force, providing the necessary details to determine how long the hose will take to unwind completely.

Angular displacement conveys how far an object has rotated, measured in radians. It is a pivotal concept in understanding the extent of rotation an object undergoes.
In the context of our scenario, the hose unwinding from the reel corresponds to an angular displacement determined by the formula \( \theta = \frac{s}{R} \), where \( s \) is the length of the hose (15.0 meters, in this case), and \( R \) is the radius of the reel (0.160 meters).
Plugging in these values, we find that the angular displacement \( \theta \) is \( 93.75 \, \text{rad} \). This value is crucial to solving for the time it takes for the reel to completely unwind the hose, using the principles of rotational kinematics.