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 and thickness of the hose on the reel, and assume that the hose unwinds without slipping.

Key Concepts

Rotational Dynamics

Rotational dynamics involves applying Newton’s second law to rotating bodies where the net torque acting on an object is equal to the product of its moment of inertia and its angular acceleration. This concept helps analyze situations where torques cause changes in rotational motion, similar to the way forces cause linear acceleration in linear dynamics.

Moment of Inertia

The moment of inertia is a measure of an object’s resistance to change in its rotational state. It depends on the mass distribution relative to the axis of rotation and plays a crucial role in determining how the applied torques will affect the angular acceleration of the object.

Net Torque

Net torque is the vector sum of all torques acting on a rotating object. In problems where multiple torques (such as an applied tension and a frictional torque) are present, calculating the net torque is essential for determining the resulting angular acceleration using the relation ?_net = I?.

Relationship Between Linear and Angular Quantities

There is a direct relationship between linear and angular quantities in rotational motion. Specifically, the linear displacement along the rim of a rotating object is related to its angular displacement through the radius, which is essential when translating a linear length (such as the length of a hose) into an angular displacement.

Rotational Kinematics

Rotational kinematics involves equations that relate angular displacement, angular velocity, angular acceleration, and time. These equations, analogous to those in linear kinematics, enable the calculation of motion parameters over time when the angular acceleration is constant.

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