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.
(a) Divide the change in angular rotation rate by the time required for the acceleration to occur.
alpha = (14 rev/s)*(2 pi rad/rev)/Time
Since 55 revolutions were required and the average rotation rate was 14 rev/s, the time required was 55/14 = 3.93 s.
alpha = 22.4 rad/s^2
(b) already determined above
(c) It will take 1/2 as long to go from 0 to 7 rev/s as it takes to go from 7 ro 21 rev/s
(d) In radians, the answer is
(angle) = (1/2) alpha t^2, where t is the time you get in part (c). Divide by 2 pi for he number of revolutions
To calculate the angular acceleration (a):
1. First, find the change in angular rotation rate: ω2 - ω1 = 21 rev/s - 7 rev/s = 14 rev/s.
2. Convert the change in angular rotation rate to radians per second: 14 rev/s * 2π rad/rev = 28π rad/s.
3. Calculate the time required for the acceleration to occur: Δt = 55 rev / 14 rev/s = 3.93 s.
4. Now, calculate the angular acceleration:
α = (Δω) / Δt = 28π rad/s / 3.93 s = 8.96 rad/s².
So, the angular acceleration is 8.96 rad/s².
To calculate the time required to complete the 55 revolutions (b):
You have already determined that the time required is 3.93 s.
To calculate the time required to attain the 7 rev/s angular speed (c):
1. Since it takes 1/2 as long to go from 0 to 7 rev/s as it takes to go from 7 to 21 rev/s, divide the time required to complete the 55 revolutions by 2: 3.93 s / 2 = 1.97 s.
Therefore, it will take 1.97 s to attain the 7 rev/s angular speed.
To calculate the number of revolutions from rest until the time the disk attained the 7 rev/s angular speed (d):
1. Use the formula for angular displacement in radians: θ = (1/2) * α * t², where α is the calculated angular acceleration, and t is the time it takes to attain the 7 rev/s angular speed, which is 1.97 s.
2. Convert the angular displacement from radians to revolutions: θ / (2π rad/rev).
By plugging in the values, you can find the number of revolutions from rest until the time the disk attained the 7 rev/s angular speed.