A 15.0 -m length of hose is wound around a reel, which is...

A $15.0$ -m length of hose is wound around a reel, which is initially at rest. The moment of inertia of the reel is $0.44 \mathrm{~kg} \cdot \mathrm{m}^{2}$, and its radius is $0.160 \mathrm{~m}$. When the reel is turning, friction at the axle exerts a torque of magnitude $3.40 \mathrm{~N} \cdot \mathrm{m}$ on the reel. If the hose is pulled so that the tension in it remains a constant $25.0 \mathrm{~N}$, how long does it take to completely unwind the hose from the reel? Neglect the mass of the hose, and assume that the hose unwinds without slipping.

Key Concepts

Relationship Between Linear and Angular Quantities

This concept explains how linear measures such as displacement, velocity, and acceleration can be related to their angular counterparts via the radius of rotation. For instance, the length of material unwound from a rotating object can be directly determined from the angular displacement by using the relation s = r?, thereby linking linear and rotational motions.

Newton's Second Law for Rotation

Newton's second law for rotation states that the net torque acting on a body is equal to the product of its moment of inertia and its angular acceleration (net torque = moment of inertia × angular acceleration). This fundamental principle allows us to relate the applied torques, including both driving and resistive torques, to the resulting rotational motion.

Rotational Kinematics

Rotational kinematics deals with the relationships between angular displacement, angular velocity, angular acceleration, and time. When an object rotates with constant angular acceleration, equations analogous to those in linear kinematics can be used to relate these variables. This concept is essential for predicting how long it takes for a rotating system to complete a given angular displacement.

Torque

Torque is a measure of the rotational force acting on an object and is calculated as the product of the force applied and the lever arm distance from the pivot point. In problems involving rotational motion, forces acting at a distance from the axis of rotation produce torque, which in turn causes angular acceleration. Both applied forces (like tension) and resistive forces (like friction) contribute to the net torque acting on a system.

Moment of Inertia

The moment of inertia is a property of a body that quantifies its resistance to angular acceleration about a given axis. It depends on the mass distribution relative to the axis of rotation. In rotational dynamics, it plays a role analogous to mass in linear motion and is critical for determining the response of a system to an applied net torque.

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