Circular rotational motion (kinematics)

[dunn]

Homework Statement

A wheel starts from rest and accelerates uniformly to 200rpm in 6s. After it has been rotating for some time at this speed, the brakes are applied and it takes 5 minutes to stop the wheel. If the total number of revolutions is 3100, calculate the total time of rotation.

Homework Equations

α = dω/dt
ω = dθ/dt

The Attempt at a Solution

Okay, so I need to calculate the number of rotations it completes in the first interval (between 0 and 6 seconds). I have the final angular velocity (ω) at the end of the first interval, so I need to find the angular acceleration (α) in order to calculate the number of radians (θ).

Since α is uniform, I integrate α = dω/dt and derive the following:

α(t-t₀) = ω - ω₀ (1)

Since t₀ and ω₀ = 0 at the start, I have:

α(t) = ω

At t = 6 and with ω = 8π/3 rad/s (200rpm), my angular acceleration α = 4π/9 rad/s²

Now I have ω = dθ/dt so α(t-t₀) = ω - ω₀ becomes:

α(t-t₀) + ω₀ = dθ/dt

Integrating this yields:

1/2α(t-t₀)² + ω₀(t-t₀) = θ - θ₀ (2)

Putting in the relevant values (t = 6, α = 4π/9, ω₀ = 0) I obtain θ = 8π rad or 4 rotations.

Now I need to calculate the last interval when the brakes are applied. Using equation (1) I obtain an angular acceleration α = -8π/900 rad/s².

Now I put the relevant information into equation 2 (t = 300, ω₀ = 8π/3 rad/s, α = -8π/900 rad/s² I get θ = 400π or 200 rotations.

Since the wheel rotates 3100 times in total, I subtract the rotations from the first and last interval to obtain 3100 - 200 - 4 = 2198 rotations that occur during the middle interval. Since 2198 rotations = 5792 radians and ω is constant, I can integrate ω = dθ/dt to obtain:

ω(t-t₀) = θ - θ₀

or, more relevantly:

ωt = θ

Since I know ω and θ I divide 5792 rad by 8π/3 rad/s and get 691.4 seconds. Adding the times from the other intervals (6s and 300s) I get a total time of 997.4 seconds or ~16.62 minutes.

The problem is that the book says the answer should be 15.6 minutes. That would mean that the wheel would complete 3120 rotations in 15.6 minutes if it started at its full speed of 200rpm and never slowed down. That simply doesn't seem right.

I want to be sure that this is just an error in the book, so I'm posting this here in the hope that someone can look over my work and make sure I did everything correctly. Am I doing something wrong or is this just a typo?

 

[rl.bhat]

200 rpm = 200*2π/60 rad/s = ...?

 

[dunn]

200 rpm = 200*2π/60 rad/s = ...?

I hate calculators that can't do fractions...

Thank you!

 

[dunn]

Well, after playing with the problem more I came to the conclusion that there is in fact a typo in the book. The brakes are supposed to be applied for a total of 5 seconds, not 5 minutes. My initial math was wrong but this was further complicated by the fact that my correct math was rendering the wrong answer because the book has (yet another!) typo. I need a new book.

 

[asteriatic]

Based on your calculations, it appears that you have approached the problem correctly. However, it is possible that there is a typo in the book or that the problem is poorly worded, leading to confusion and incorrect answers. As a scientist, it is important to critically examine all sources of information and question any discrepancies or inconsistencies. In this case, it may be helpful to double check your calculations and assumptions, and also consider reaching out to the author or instructor for clarification. Overall, your approach to the problem is sound and it is likely that the discrepancy lies in the given information rather than your calculations.