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 sec
(b) What is the angular acceleration of each wheel?
in rad/sec2
(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 = 2π 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.
To find the answers to these questions, we can use the equations of rotational motion. There are three key equations we can use:
1. Angular displacement (θ) = Angular velocity (ω) * Time (t) + 0.5 * Angular acceleration (α) * (Time (t))^2
2. Angular velocity (ω) = Initial angular velocity (ω0) + Angular acceleration (α) * Time (t)
3. Angular velocity squared (ω^2) = Initial angular velocity squared (ω0^2) + 2 * Angular acceleration (α) * Angular displacement (θ)
Let's begin with part (a):
(a) How much time does it take for the bike to come to rest?
We are given that the angular displacement of each wheel is +15.5 revolutions and the angular velocity is +15.5 rad/s. We need to find the time it takes for the bike to stop.
Using equation (1), we can rearrange it to solve for time (t):
θ = ω0t + 0.5αt^2
We know that θ = 15.5 revolutions and ω0 = +15.5 rad/s. Since 15.5 revolutions is equivalent to 2π * 15.5 rad, we can convert the angular displacement from revolutions to radians.
15.5 revolutions * 2π rad/revolution = 97.4π rad
Now, we can plug in the values into the equation and solve for time (t):
97.4π rad = (15.5 rad/s)t + 0.5αt^2
Simplifying the equation:
0.5αt^2 + (15.5 rad/s)t - 97.4π rad = 0
Now we have a quadratic equation in terms of t. We can solve it using the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a)
Where a = 0.5α, b = 15.5 rad/s, and c = -97.4π rad.
After solving the quadratic equation, you will get two possible values for time (t). Keep in mind that we are only interested in the positive solution since time cannot be negative. This positive value will be the time it takes for the bike to come to rest.
Now let's move to part (b):
(b) What is the angular acceleration of each wheel?
We already know the initial angular velocity (ω0) and the angular displacement (θ) of each wheel.
Using equation (3), we can rearrange it to solve for angular acceleration (α):
ω^2 = ω0^2 + 2αθ
Substituting the given values:
(0 rad/s)^2 = (15.5 rad/s)^2 + 2α * 15.5 revolutions
(0 rad/s)^2 = (15.5 rad/s)^2 + 2α * 2π * 15.5 rad
Simplifying the equation:
0 = (15.5 rad/s)^2 + 2πα * 15.5 rad
We can now solve for angular acceleration (α) by rearranging the equation and dividing by 2π * 15.5 rad:
α = -(15.5 rad/s)^2 / (2π * 15.5 rad)
Simplifying further:
α = -(15.5 rad/s) / (2π)
After performing the calculation, you'll find the value of the angular acceleration (α) in rad/sec^2.
Overall, by using the equations of rotational motion and solving for the respective variables, you can find the answers to these questions.