A 13.6-m length of hose is wound around a reel, which is initially at rest. The moment of inertia of the reel is 0.40 kg · m2, and its radius is 0.151 m. When the reel is turning, friction at the axle exerts a torque of magnitude 3.42 N · m on the reel. If the hose is pulled so that the tension in it remains a constant 23.1 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.

Answer in seconds

To determine the time it takes to completely unwind the hose from the reel, we can start by calculating the angular acceleration of the reel using the torque equation.

The torque exerted by friction at the axle can be given by:

τ = I * α

Where:
τ = Torque (3.42 N · m)
I = Moment of inertia of the reel (0.40 kg · m^2)
α = Angular acceleration

Rearranging the equation to solve for α:

α = τ / I

Plugging in the values:

α = 3.42 N · m / 0.40 kg · m^2
= 8.55 rad/s^2

Now, we need to find the angular velocity of the reel when the hose is completely unwound. The angular velocity can be calculated using the equation:

ω^2 = ω0^2 + 2αθ

Where:
ω = Angular velocity
ω0 = Initial angular velocity (assumed to be 0 as the reel is initially at rest)
α = Angular acceleration (8.55 rad/s^2)
θ = Angle (in radians) through which the hose is unspooled (2π, as the hose is completely unwound)

Rearranging the equation to solve for ω:

ω = sqrt(ω0^2 + 2αθ)
= sqrt(0 + 2(8.55 rad/s^2)(2π))
= 5.47 rad/s

Finally, we can determine the time it takes to completely unwind the hose by dividing the total angle through which the hose is unspooled (2π) by the angular velocity:

Time = θ / ω
= 2π / 5.47 rad/s
≈ 1.14 seconds

Therefore, it takes approximately 1.14 seconds to completely unwind the hose from the reel.

To determine how long it takes to completely unwind the hose from the reel, we need to use the principles of rotational dynamics and torque. Here's how to approach the problem step by step:

1. First, let's start by finding the angular acceleration of the reel. The torque exerted by the friction force can be calculated using the equation:

τ = Iα

Where τ is the torque, I is the moment of inertia, and α is the angular acceleration.

Given: τ = 3.42 N·m, I = 0.40 kg·m^2

We can rearrange the equation to solve for α:

α = τ / I

α = 3.42 N·m / 0.40 kg·m^2

Calculating α gives us the angular acceleration.

2. Now, we can use the tension in the hose to determine the torque produced by it. The tension creates a torque because it acts at a distance from the center of the reel. The equation for torque due to tension is:

τ = r*T

Where τ is the torque, r is the radius, and T is the tension.

Given: r = 0.151 m, T = 23.1 N

Calculating τ gives us the torque produced by the tension.

3. Since there is no external torque acting on the system besides the friction torque and the tension torque, we can equate the two torques to find the net torque acting on the reel:

τ(net) = τ(friction) + τ(tension)

4. Once we have the net torque, we can now determine the angular acceleration using the equation:

τ(net) = I*α

Rearranging the equation allows us to solve for α.

5. Now that we have the angular acceleration, we can apply the rotational kinematic equation to find the time it takes for the hose to completely unwind from the reel:

θ = ω0*t + (1/2)*α*t^2

Assuming the reel starts from rest, ω0 is zero. Since the hose needs to unwind completely, the angle θ is 2π radians.

Rearranging the equation will allow us to solve for t.

Now that you have the step-by-step explanation, you can calculate the answer using the given values and the equations provided.