A 15.5-m length of hose is wound around a reel, which is initially at rest. The moment of inertia of the reel is 0.42 kg · m2, and its radius is 0.170 m. When the reel is turning, friction at the axle exerts a torque of magnitude 3.63 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.

Hi, for this one you will be doing

Applied torque minus frictional torque
(23.1)(.17)-3.63=.28 N-m
(summarizing where I got the numbers above from the problem;)
(remains constant at)(radius)-(torque w/ magnitude)

then you want to find the angular acceleration. Ac=T/ I (Torque over moment of intertia)
We know the torque is now .28 N-m and the moment of inertia from the problem is .42 so
(.28)/(.42)=.66 or .67 radians/s ^2 rounded for the angular acceleration.

Next we will use the hose length/ radius so 15.5/.17=91.1 radians

Finally to find the answer to our problem we will do;

91.1 radians=(1/2) (.67)t^2

We are solving for t. so
91.1=.355t^2

divide by .355 on each side
256.62=t^2
then square root to isolate t to get
16.02 s

I hope you can apply this strategy to the numbers on your problem that you get . Have fun with Physics :)

The applied torque minus the frictional torque is (23.1)(0.170)-3.63 = 0.28 N-m

Torque divided by moment of inertia is the angular acceleration, alpha. Its units are radians/s^2. Calculate it

To unreel all of the hose, you need to turn the reel through 15.5/0.17 = 91.1 radians.

Solve this equation for the required time, t:

91.1 radians = (1/2)*(alpha)*t^2

Hmm, let's unwind this problem, shall we? We need to find out how long it takes to completely unwind the hose from the reel.

To do that, we need to consider the torque exerted on the reel. The torque exerted by friction is given as 3.63 N · m.

Now, we know that torque is equal to the moment of inertia (I) multiplied by the angular acceleration (α). Since the reel starts from rest, the initial angular velocity (ω) is 0.

So, the torque can be written as τ = I * α. Rearranging the equation, we can find the angular acceleration:

α = τ / I.

Plugging in the given values, α = 3.63 N · m / 0.42 kg · m^2.

Now, we need to find the angular velocity (ω) when the hose is completely unwound. We can use the equation:

ω^2 = ω0^2 + 2αθ,

where ω0 is the initial angular velocity (0 in this case) and θ is the angle through which the hose has been unwound (360 degrees or 2π radians).

Let's calculate α first:

α = 3.63 N · m / 0.42 kg · m^2 = 8.64 rad/s^2.

Now, plugging the values into the equation for angular velocity:

ω^2 = 0^2 + 2 * 8.64 rad/s^2 * 2π rad.

Simplifying it further:

ω^2 = 108.38 rad^2/s^2.

Finally, taking square root on both sides to find ω:

ω = √(108.38 rad^2/s^2).

Now, we know that the angular velocity (ω) is given by ω = θ / t, where t is the time in seconds.

Rearranging the equation, we get:

t = θ / ω.

Since the hose needs to be completely unwound (θ = 2π rad), we can plug in the values:

t = 2π rad / √(108.38 rad^2/s^2).

Calculating it, we find:

t ≈ 1.168 seconds.

So, it takes approximately 1.168 seconds to completely unwind the hose from the reel. Keep those hoses rolling!

To solve this problem, we can use the principles of rotational motion and torque.

Step 1: Find the net torque acting on the reel.
The net torque is the difference between the applied torque and the friction torque. Given that the frictional torque is 3.63 N · m and assuming that the applied torque is in the opposite direction, the net torque is -3.63 N · m.

Step 2: Find the angular acceleration of the reel.
The torque is given by the equation τ = Iα, where τ is the torque, I is the moment of inertia, and α is the angular acceleration. Rearranging the equation, we have α = τ / I.

In this case, τ = -3.63 N · m and I = 0.42 kg · m^2. Substituting these values, α = -3.63 N · m / 0.42 kg · m^2 = -8.64 rad/s^2.

Step 3: Find the angular velocity of the reel.
The angular velocity can be determined using the equation ω = ω₀ + αt, where ω is the final angular velocity, ω₀ is the initial angular velocity (which is zero in this case since the reel is initially at rest), α is the angular acceleration, and t is the time.

Since the initial angular velocity is zero, the equation simplifies to ω = αt. Substituting the values, ω = -8.64 rad/s^2 x t.

Step 4: Find the time it takes to completely unwind the hose.
The length of the hose wound around the reel represents the distance traveled by a point on the circumference of the reel. The distance traveled is equal to the angular displacement multiplied by the radius.

The total length of the hose is 15.5 m, which is equal to the circumference of the reel (2πr). Using this information, we can set up the equation:

15.5 m = (2πr) × θ

To unwind the hose completely, θ must be equal to 2π (one full revolution). Rearranging the equation to solve for θ:

θ = 15.5 m / (2πr).

Substituting the given radius value of 0.17 m, θ = 15.5 m / (2π × 0.17 m) = 45.98 rad.

Now, we can set up the final equation to find the time:

45.98 rad = -8.64 rad/s^2 x t.

Solving for t, t = 45.98 rad / -8.64 rad/s^2 ≈ -5.33 s.

Since time cannot be negative in this context, the time it takes to completely unwind the hose from the reel is approximately 5.33 seconds.

To determine how long it takes to completely unwind the hose from the reel, we need to find the angular acceleration of the reel first.

1. Start by finding the net torque acting on the reel. The torque exerted by the friction at the axle is given as 3.63 N · m.

2. The tension in the hose also creates a torque when it is pulled. The torque created by tension is given by the equation:
Tension * radius of the reel = Torque
Substituting the given values, we have:
23.1 N * 0.170 m = Torque

3. The net torque is the difference between the torque created by friction and the torque created by tension:
Net torque = Torque from tension - Torque from friction

4. Now, we can use the equation for torque and angular acceleration to find the angular acceleration:
Torque = moment of inertia * angular acceleration
Rearranging the equation:
Angular acceleration = Torque / moment of inertia

5. Substitute the values for the torque and moment of inertia into the equation to find the angular acceleration.

6. Once we have the angular acceleration, we can use it to find the time it takes to completely unwind the hose using the equation:
Angular acceleration = change in angular velocity / time taken

In this case, the initial angular velocity is 0 since the reel is initially at rest. The final angular velocity is the angular velocity when the hose is completely unwound, which is what we are trying to find. Rearranging the equation:
Time taken = Final angular velocity / Angular acceleration

7. Finally, substitute the calculated values of the angular acceleration and solve for the time taken to completely unwind the hose from the reel.