4radEach revolution has 2π radians (that is, 2π rad/1 rev = 1)3023.4rad / (2π rad/1 rev)= 3023.4rev / 2π= 481.19 revolutions--> 481 revolutions-Break the problem up into three phases: (1) acceleration phase; (2) constant angular speed phase; (3) deceleration phase.Find the number of revolutions in each phase and then add them up.......
ωf^2 - ωi^2 = 2αθ
=> θ = (ωf^2 - ωi^2)/2α
θ = [(0rad/s)^2 - (53rad/s)^2] / 2(-2.2rad/s^2)
θ = (- 2809rad^2/s^2) / (-4.4rad/s^s)
θ = 638.4rad
Now, add them all together
θ(total) = 638.4rad + 2120rad + 265rad
θ(total) = 3023.4rad
Each revolution has 2π radians (that is, 2π rad/1 rev = 1)
3023.4rad / (2π rad/1 rev)
= 3023.4rev / 2π
= 481.19 revolutions
--> 481 revolutions
Break the problem up into three phases: (1) acceleration phase; (2) constant angular speed phase; (3) deceleration phase. Find the number of revolutions in each phase and then add them up.
(Actually, to save a math step, we'll add up the number of RADIANS rotated in each phase, and convert to revolutions at the end. Same answer, fewer steps.)
Phase 1:
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We're given:
1. Initial angular speed: ωi = 0
2. Final angular speed: ωf = 53 rad/s
3. Elapsed time: t = 10s
Use the kinematics equation that relates these three quantities with the total rotation in radians (we'll call that "θ1". (The "1" stands for "Phase 1"))
θ1 = ½(ωi + ωf)(t)
(Note: This formula only works when the acceleration is uniform (which it is, in this problem)).
Phase 2:
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We're given:
1. Constant angular speed: ω = 53 rad/s
2. Elapsed time: t = 40s
(Note: this "t" is not the same "t" from Phase 1; I'm recycling the variable name.)
Use the standard kinematics equation relating constant angular speed, time, and radians:
θ2 = (ω)(t)
Phase 3:
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We're given:
1. Initial angular speed: ωi = 53 rad/s
2. Final angular speed: ωf = 0
3. Angular acceleration: α = −2.2 rad/s² (negative because it's slowing down)
(Note: These "ωi" and "ωf" are not the same as the variables in Phase 1. I'm recycling the variable names.)
Use the kinematics equation that relates these three quantities with θ3:
ωf² − ωi² = 2α(θ3)
Final Step:
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Add up θ1 + θ2 + θ3 to get the total number of radians rotated; then divide that by 2π to convert from radians to revolutions.
