A person is riding a bicycle, the wheels of a bicycle have an angular velocity of +20.5 rad/s. Then, the brakes are applied. In coming to rest, each wheel makes an angular displacement of +12.0 revolutions.
(a) How much time does it take for the bike to come to rest?
(b) What is the angular acceleration of each wheel?
To find the time it takes for the bike to come to rest, we can use the fact that the angular displacement is related to the angular velocity and time through the equation:
θ = ω₀t + (1/2)αt²
where θ is the angular displacement in radians, ω₀ is the initial angular velocity in rad/s, α is the angular acceleration in rad/s², and t is the time in seconds.
(a) Given that each wheel makes an angular displacement of +12.0 revolutions, we first need to convert it to radians. Since 1 revolution is equal to 2π radians, the angular displacement is:
θ = 12.0 revolutions * (2π radians/1 revolution) = 24π radians
We are also given the initial angular velocity ω₀ = +20.5 rad/s.
To find the time t, we rearrange the equation and solve for t:
θ = ω₀t + (1/2)αt²
Substituting the given values, we have:
24π = (20.5)t + (1/2)αt²
Now, we need another piece of information to solve for t. We are given the angular velocity, but we need the angular acceleration to find the time.
To find the angular acceleration α, we use the equation:
ω = ω₀ + αt
where ω is the final angular velocity, which is 0 (since the bike comes to rest), and t is the time.
Substituting the given values, we have:
0 = 20.5 + αt
Solving for α in terms of t:
αt = -20.5
α = -20.5/t
Now, we can substitute this expression for α in the equation for θ:
24π = (20.5)t + (1/2)(-20.5/t)t²
Simplifying the equation:
24π = 20.5t - 10.25t
Combining like terms:
24π = 10.25t
Dividing both sides by 10.25:
t = 24π/10.25
Using a calculator, we get:
t ≈ 7.40 seconds
So, it takes approximately 7.40 seconds for the bike to come to rest.
(b) Now that we have found the time, we can find the angular acceleration α using the equation:
α = -20.5/t
Substituting the value of t we found:
α = -20.5/7.40
Using a calculator, we get:
α ≈ -2.77 rad/s²
Therefore, the angular acceleration of each wheel is approximately -2.77 rad/s². Note: The negative sign indicates the deceleration of the bike.