(a) To find the time it takes for the bike to come to rest, we need to calculate the angular acceleration first. Then, we can use the equation of motion for angular acceleration to find the time.
(b) Given:
Angular velocity, ω = +15.5 rad/s
Angular displacement, θ = +15.5 revolutions
To convert revolutions to radians, we can use the conversion factor: 1 revolution = 2Ï€ radians.
Therefore, θ = +15.5 revolutions = +15.5 * 2π radians.
Now, we know that the equation of motion for angular acceleration is:
ω^2 = ω0^2 + 2αθ
where
ω = final angular velocity (0 rad/s, as the bike comes to rest)
ω0 = initial angular velocity (15.5 rad/s)
α = angular acceleration
θ = angular displacement
Plugging in the given values, we can solve for α:
0^2 = (15.5 rad/s)^2 + 2 * α * (+15.5 * 2π radians)
0 = 240.25 rad^2/s^2 + 2α * 15.5 * 2π radians
Now, solve for α:
2α * 15.5 * 2π radians = -240.25 rad^2/s^2
α = (-240.25 rad^2/s^2) / (2 * 15.5 * 2π radians)
Simplifying:
α = -240.25 / (31 * π) rad/s^2
Now that we have α, we can find the time it takes for the bike to come to rest using the equation:
ω = ω0 + α * t
0 = 15.5 rad/s + (-240.25 / (31 * π)) rad/s^2 * t
-15.5 rad/s = (-240.25 / (31 * π)) rad/s^2 * t
Simplifying:
t = (-15.5 rad/s) / (-240.25 / (31 * π)) rad/s^2
t = (15.5 rad/s) * (31 * π) / 240.25 rad/s^2
Calculating the value:
t ≈ 6.272 seconds
Therefore, it takes approximately 6.272 seconds for the bike to come to rest.
(b) The angular acceleration of each wheel is approximately -240.25 / (31 * π) rad/s^2.