A person is riding a bicycle, the wheels of a bicycle have an angular velocity of +18.5 rad/s. Then, the brakes are applied. In coming to rest, each wheel makes an angular displacement of +14.5 revolutions.
(a) How much time does it take for the bike to come to rest?
1 s
(b) What is the angular acceleration of each wheel?
2 rad/s2
To calculate the time it takes for the bike to come to rest, we can use the formula:
Angular displacement = Angular velocity * Time + 0.5 * Angular acceleration * Time^2
In this case, the angular displacement is given as +14.5 revolutions, which can be converted to radians:
14.5 revolutions x 2π rad/revolution = 14.5 x 2π radians
The angular velocity is given as +18.5 rad/s, and the final angular velocity will be 0 since the bike comes to rest.
Plugging in these values to the formula, we have:
14.5 x 2π = 18.5 * t + 0.5 * a * t^2
Simplifying the equation, we get:
29π = 18.5t + 0.5at^2
Since the question does not provide information on the angular acceleration, we assume it to be constant.
To find the value of angular acceleration, we can differentiate the equation with respect to time (t):
0 = 18.5 + at
Simplifying this equation, we get:
at = -18.5
Now, we can substitute the value of 'a' as -18.5 into the first equation:
29π = 18.5t - 9.25t^2
Simplifying further, we have:
9.25t^2 - 18.5t + 29π = 0
Using the quadratic formula, t = (-b +/- sqrt(b^2 - 4ac))/(2a), where a = 9.25, b = -18.5, and c = 29π, we find:
t = (-(-18.5) +/- sqrt((-18.5)^2 - 4*9.25*29π))/(2*9.25)
Simplifying this equation, we get:
t = (18.5 +/- sqrt(342.25 - 265.75π))/18.5
Since we are interested in the positive value for time, let's calculate the value of t:
t ≈ 1 s
So, the time it takes for the bike to come to rest is approximately 1 second.
To find the angular acceleration of each wheel, we can substitute the value of t = 1 second into the equation:
at = -18.5
a = -18.5/1
a = -18.5 rad/s^2
Since angular acceleration is a scalar quantity, we take the magnitude of the negative value:
|a| = 18.5 rad/s^2
Therefore, the angular acceleration of each wheel is approximately 18.5 rad/s^2.
To find the time it takes for the bike to come to rest, we need to use the equation relating angular displacement, angular velocity, and angular acceleration:
θ = ω₀t + 0.5αt²
Where:
θ = angular displacement (in radians)
ω₀ = initial angular velocity (in rad/s)
α = angular acceleration (in rad/s²)
t = time (in seconds)
For each wheel, let's convert the given angular displacement from revolutions to radians:
θ = 14.5 revolutions * 2π radians/1 revolution
θ = 14.5 * 2π radians
Now, we know that the final angular velocity is 0 rad/s (because the bike comes to rest), and the initial angular velocity (ω₀) is +18.5 rad/s. So, we have:
θ = ω₀t + 0.5αt²
0 = (+18.5 rad/s)t + 0.5αt²
Since we are looking for the time it takes for the bike to come to rest, we can set θ equal to the given displacement:
14.5 * 2π = (+18.5 rad/s)t + 0.5αt²
Using this equation, we can solve for the time (t). However, we don't have enough information to find the angular acceleration (α) directly from this equation. We can only determine the angular acceleration if we have additional information or if it is given.
Since we don't know the angular acceleration, we cannot find the time or angular acceleration in this case.