A disk rotates about its central axis starting from rest and accelerates with constant angular acceleration. At one time it is rotating at 7 rev/s. 55 revolutions later, its angular speed is 21 rev/s.
(a) Calculate the angular acceleration.
(b) Calculate the time required to complete the 55 revolutions mentioned.
(c) Calculate the time required to attain the 7 rev/s angular speed.
(d) Calculate the number of revolutions from rest until the time the disk attained the 7 rev/s angular speed.
This is pretty straighforward. What is about it that you do not understand?
what equations do I use? thanks for your help.
To solve this problem, you can use the equations of rotational motion. Here are the equations you will need:
1. Angular velocity: ω = ω0 + αt, where ω is the final angular velocity, ω0 is the initial angular velocity, α is the angular acceleration, and t is the time.
2. Angular displacement: θ = ω0t + 0.5αt^2, where θ is the angular displacement, ω0 is the initial angular velocity, α is the angular acceleration, and t is the time.
3. Number of revolutions: N = θ / (2π), where N is the number of revolutions and θ is the angular displacement.
Now let's solve the problem step by step:
(a) To calculate the angular acceleration (α), you can use the equation ω = ω0 + αt. We are given that the initial angular velocity (ω0) is 0 rev/s and the final angular velocity (ω) is 21 rev/s. Since the disk starts from rest and accelerates, ω0 is 0. Plugging in the values, we get:
ω = ω0 + αt
21 = 0 + αt
Solving for α, we have:
α = ω / t
α = 21 rev/s / t
(b) To calculate the time required to complete the 55 revolutions mentioned, you can use the equation N = θ / (2π). We are given that N (number of revolutions) is 55. Plugging in the values, we get:
N = θ / (2π)
55 = θ / (2π)
Solving for θ, we have:
θ = N * 2π
θ = 55 * 2π
Now we can calculate the time using θ and the equation θ = ω0t + 0.5αt^2. Rearranging the equation, we have:
0.5αt^2 + ω0t - θ = 0
Substituting the values, we have:
0.5 * (21 rev/s) * t^2 + 0 * t - (55 * 2π) = 0
Using quadratic formula, we can solve for t.
(c) To calculate the time required to attain the 7 rev/s angular speed, you can use the equation ω = ω0 + αt. We are given that the initial angular velocity (ω0) is 0 rev/s and the final angular velocity (ω) is 7 rev/s. Plugging in the values, we get:
ω = ω0 + αt
7 = 0 + αt
Solving for α, we have:
α = ω / t
α = 7 rev/s / t
(d) To calculate the number of revolutions from rest until the time the disk attained the 7 rev/s angular speed, you can use the equation N = θ / (2π). We are given that the initial angular velocity (ω0) is 0 rev/s and the final angular velocity (ω) is 7 rev/s. Rearranging the equation ω = ω0 + αt, we have:
t = ω / α
t = 7 rev/s / (7 rev/s / t)
Now we can calculate the time using t and the equation θ = ω0t + 0.5αt^2. Rearranging the equation, we have:
θ = 0 * t + 0.5αt^2
θ = 0.5αt^2
Substituting the values, we have:
θ = 0.5 * (7 rev/s / t) * t^2
Now, you can solve for θ by plugging in the values and simplifying the equation.
I hope this helps! Let me know if you have any more questions.