Search: A person is riding a bicycle, the wheels of a bicycle have an angular velocity of +18.0 rad/s. Then, the brakes are applied. In coming to rest, each wheel makes an angular displacement of +11.0 revolutions.
How much time does it take for the bike to come to rest?
What is the angular acceleration of each wheel?
Time to stop = (angular displacement)/(average angular velocity)
= 11*2*pi rad/(9.0 rad/s) = 7.68 s
Angular acceleration
= -(18 rad/s)/(7.68 s)
= -2.34 rad/s^2
We can solve these questions by using the equations of rotational motion.
First, let's find the angular acceleration of each wheel.
Angular acceleration (α) is defined as the change in angular velocity (ω) divided by the change in time (t):
α = (ωf - ωi) / t
Here, the initial angular velocity (ωi) is +18.0 rad/s, and the final angular velocity (ωf) is 0 rad/s (since the bike comes to rest).
Substituting the values into the equation, we have:
0 = (0 - 18.0) / t
Simplifying this equation, we get:
0 = -18.0 / t
Rearranging the equation, we have:
18.0 = 0 * t
Since any number multiplied by 0 is 0, we find that the angular acceleration is 0 rad/s². This means that the wheels decelerate uniformly and come to rest without changing their angular velocity.
Now, let's calculate the time it takes for the bike to come to rest.
We know that the angular displacement (θ) is +11.0 revolutions. Since 1 revolution is equal to 2π radians, the angular displacement can be converted to radians as follows:
θ = +11.0 revolutions * 2π radians/revolution
θ = +11.0 * 2π rad
θ = +22π rad
Angular displacement (θ) can be related to angular velocity (ω), angular acceleration (α), and time (t) through the following equation:
θ = ωi*t + (1/2) * α * t^2
Since α = 0 rad/s², the equation simplifies to:
θ = ωi*t
We can solve this equation for time (t):
t = θ / ωi
Substituting the given values:
t = (+22π rad) / (+18.0 rad/s)
Simplifying the expression:
t ≈ 3.85 s
Therefore, it takes approximately 3.85 seconds for the bike to come to rest.
To find the time it takes for the bike to come to rest, we need to use the relationship between angular displacement, angular velocity, angular acceleration, and time.
The angular displacement of each wheel is given as +11.0 revolutions. Since 1 revolution is equal to 2π radians, the angular displacement can be converted to radians:
Angular displacement in radians = 11.0 revolutions * 2π radians/revolution
Now that we have the angular displacement in radians, we can find the time using the equation:
Angular displacement = (Angular initial velocity * time) + (0.5 * Angular acceleration * time^2)
Since we are interested in finding the time, we rearrange the equation:
0 = 0.5 * Angular acceleration * time^2 + (Angular initial velocity * time - Angular displacement)
Now we can substitute the given values into the equation. The initial angular velocity of each wheel is +18.0 rad/s, and the angular displacement in radians is +11.0 revolutions * 2π radians/revolution.
0 = 0.5 * Angular acceleration * time^2 + (18.0 rad/s * time - 11.0 revolutions * 2π radians/revolution)
To solve for time, we can use numerical methods or numerical software. One common approach is to use the Newton-Raphson method or a graphing calculator.
Now let's move on to finding the angular acceleration of each wheel.
The angular acceleration can be determined by using the equation:
Angular acceleration = (Final angular velocity - Initial angular velocity) / Time
Since the bike comes to rest, the final angular velocity is 0. Therefore, we can simplify the equation to:
Angular acceleration = -Initial angular velocity / Time
Substituting the given values, the initial angular velocity is +18.0 rad/s, and the time can be found using the previous calculation.
Angular acceleration = -18.0 rad/s / (time calculated from previous step)
Now that you have the equation and the numerical values, you can calculate the time it takes for the bike to come to rest and the angular acceleration of each wheel.