A person is riding a bicycle, and its wheels have an angular velocity of 28.9 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 13.6 revolutions. (a) How much time does it take for the bike to come to rest? (b) What is the anguar acceleration (in rad/s2) of each wheel?
followed this format and it was wrong!!
writing the correct way to help others:
rev *6.28 = rads
t= 2 *rads/angular velocity
a = angular velocity/2*rads = - (Answer)
To find the time it takes for the bike to come to a stop, we can use the formula:
ω = Δθ / Δt
where ω is the angular velocity, Δθ is the angular displacement, and Δt is the time it takes.
(a) We are given that the angular velocity of each wheel is 28.9 rad/s, and the angular displacement of each wheel is 13.6 revolutions.
Converting the angular displacement from revolutions to radians:
13.6 revolutions * 2π radians/revolution = 13.6 * 2π radians ≈ 85.942 radians
Now, we can rewrite the formula as:
28.9 rad/s = 85.942 radians / Δt
Solving for Δt:
Δt = 85.942 radians / 28.9 rad/s ≈ 2.974 seconds
Therefore, it takes approximately 2.974 seconds for the bike to come to rest.
(b) To find the angular acceleration, we can use the formula:
α = Δω / Δt
where α is the angular acceleration, Δω is the change in angular velocity, and Δt is the time it takes.
Since the bike comes to a stop, the final angular velocity is 0 rad/s and the initial angular velocity is 28.9 rad/s.
Substituting the values:
α = (0 rad/s - 28.9 rad/s) / 2.974 seconds ≈ -9.713 rad/s^2
The angular acceleration of each wheel is approximately -9.713 rad/s^2. (Note: the negative sign indicates deceleration.)
To answer these questions, we can use the formulas related to angular velocity, angular displacement, and angular acceleration.
(a) To find the time it takes for the bike to come to rest, we can use the formula:
time = angular displacement / angular velocity
Given:
Angular velocity (ω) = 28.9 rad/s
Angular displacement (θ) = 13.6 revolutions
First, let's convert the angular displacement from revolutions to radians:
1 revolution = 2π radians
So, 13.6 revolutions = 13.6 * 2π radians
Plug the values into the formula:
time = (13.6 * 2π radians) / 28.9 rad/s
Now we can calculate the time it takes for the bike to come to rest.
(b) To find the angular acceleration of each wheel, we can use the formula:
angular acceleration = change in angular velocity / time
Since the bike comes to rest, the final angular velocity is 0 rad/s.
Therefore, change in angular velocity (Δω) = final angular velocity - initial angular velocity = 0 - 28.9 rad/s = -28.9 rad/s
Now, we can use the formula to calculate the angular acceleration:
angular acceleration = (-28.9 rad/s) / time (from part a)
By plugging the value of time obtained in part a, you can calculate the angular acceleration of each wheel in rad/s².
a. 28.9rad/s * 1rev/6.28rad = 4.60 rev/s.
t = (13.6/4.60) * 1s = 3.0 s.
b. V = Vo + at = 0 When @ rest.
28.9 + a*3 = 0
3a = -28.9
a = -9.63 rad/s^2.