A person is riding a bicycle, and its wheels have an angular velocity of +15.5 rad/s. Then, the brakes are applied and the bike is brought to a uniform stop. During braking, the angular displacement of each wheel is +15.5 revolutions.
(a) How much time does it take for the bike to come to rest?
in ses
(b) What is the angular acceleration of each wheel?
in rad/sec2
To find the time it takes for the bike to come to rest, we need to use the equation of angular velocity:
ω = Δθ / t
where:
ω is the angular velocity (in radians per second),
Δθ is the angular displacement (in radians),
and t is the time (in seconds).
For part (a), we are given the angular velocity (ω) as +15.5 rad/s and the angular displacement (Δθ) as +15.5 revolutions. We need to convert the angular displacement from revolutions to radians using the conversion factor:
1 revolution = 2π radians
So, Δθ = 15.5 revolutions * 2π radians/revolution = 15.5 * 2π radians.
Now, we can rearrange the equation to solve for t:
t = Δθ / ω
Substituting the values, we get:
t = (15.5 * 2π) radians / 15.5 rad/s
The angular displacement cancels out, leaving us with:
t = 2π seconds
Therefore, it takes 2π seconds for the bike to come to rest.
For part (b), to calculate the angular acceleration (α), we can use the equation:
α = (Δω) / t
where:
α is the angular acceleration (in radians per second squared),
Δω is the change in angular velocity (in radians per second), and
t is the time (in seconds).
Since the bike is brought to a uniform stop, its final angular velocity (ωf) is 0 (zero). Thus, the change in angular velocity (Δω) is equal to the initial angular velocity (ωi) which is +15.5 rad/s.
Substituting the values, we get:
α = (+15.5 rad/s) / (2π seconds)
Simplifying, we have:
α = 15.5 / (2π) rad/s²
Therefore, the angular acceleration of each wheel is 15.5 / (2π) rad/s².
To find the time it takes for the bike to come to rest, we need to use the equation:
Δθ = ω(initial) * t + (1/2) * α * t^2
where:
Δθ = angular displacement
ω(initial) = initial angular velocity
α = angular acceleration
t = time
Given:
Δθ = +15.5 revolutions
ω(initial) = +15.5 rad/s
We need to convert the angular displacement from revolutions to radians:
1 revolution = 2π radians
So, Δθ = 15.5 revolutions * 2π radians/revolution = 31π radians
Substituting the values into the equation, we get:
31π = 15.5 * t + (1/2) * α * t^2
Now, let's find the angular acceleration (α):
The final angular velocity is 0 since the bike comes to a stop, so we have:
ω(final) = 0 rad/s
Using the equation ω(final) = ω(initial) + α * t, we can solve for α:
0 = 15.5 + α * t
Now we have two equations with two unknowns:
31π = 15.5 * t + (1/2) * α * t^2
0 = 15.5 + α * t
To solve for t and α, we can substitute the second equation into the first equation to eliminate α.
31π = 15.5 * t + (1/2) * (-15.5/t) * t^2
31π = 15.5 * t - 7.75 * t
31π = 7.75 * t
t = 31π / 7.75
Now, let's calculate the time it takes for the bike to come to rest:
t = 31π / 7.75
t ≈ 12.35 seconds
Therefore, it takes approximately 12.35 seconds for the bike to come to rest.
To find the angular acceleration (α), we can substitute the value of t into the second equation:
0 = 15.5 + α * (31π / 7.75)
-15.5 = α * (31π / 7.75)
α = -15.5 * (7.75 / (31π))
α ≈ -0.625 rad/s^2
Therefore, the angular acceleration of each wheel is approximately -0.625 rad/s^2.