A person is riding a bicycle, and its wheels have an angular velocity of 17.7 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 12.4 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?
w = w0 - a*t
theta = theta0 + w0*t - 1/2*a*t^2
where w is the angular velocity as a function of time, w0 is the initial velocity, t is time, a is the angular acceleration, theta is the angular displacement, and theta0 is the initial angular displacement.
12.4 revolutions = 2*pi*12.4 radians
when the bike stops, w = 0. Inserting values into the equation:
0 = 17.7 - a*t
2*pi*12.4 = 17.7*t - .5*a*t^2
Use algebra to solve this system of equations to find the time t that it takes the bike to come to rest, and the angular acceleration a.
= 17.7
To solve this problem, we can use the formula relating angular velocity, angular displacement, and time:
angular velocity = angular displacement / time
Let's start by converting the angular displacement from revolutions to radians:
12.4 revolutions * 2π rad/revolution = 24.8π rad
Now, we can rearrange the formula to solve for time:
time = angular displacement / angular velocity
(a) Substituting the values,
time = 24.8π rad / 17.7 rad/s
Using a calculator,
time ≈ 4.431 seconds
So, it takes approximately 4.431 seconds for the bike to come to rest.
(b) To find the angular acceleration, we can use another formula relating angular acceleration, angular velocity, and time:
angular acceleration = (final angular velocity - initial angular velocity) / time
Since the bike comes to a uniform stop, the final angular velocity is 0 rad/s. The initial angular velocity is given as 17.7 rad/s.
Substituting the values:
angular acceleration = (0 rad/s - 17.7 rad/s) / 4.431 s
Using a calculator,
angular acceleration ≈ -3.993 rad/s^2
Therefore, the angular acceleration of each wheel is approximately -3.993 rad/s^2. Note that the negative sign indicates deceleration in the opposite direction of forward motion.
To find the time it takes for the bike to come to rest, you need to use the relationship between angular displacement, angular velocity, and time. The formula is:
Angular displacement = angular velocity x time
In this case, the angular displacement is given as 12.4 revolutions. However, it's better to convert it to radians for consistency with the given angular velocity. Since there are 2π radians in one revolution, you can convert 12.4 revolutions to radians by multiplying it by 2π:
Angular displacement = 12.4 revolutions x 2π radians/revolution
Next, you can use the formula to solve for time:
12.4 revolutions x 2π radians/revolution = 17.7 rad/s x time
Solving for time, you get:
Time = (12.4 revolutions x 2π radians/revolution) / 17.7 rad/s
To calculate the angular acceleration of each wheel, you can use the formula:
Angular acceleration = (final angular velocity - initial angular velocity) / time
In this case, the final angular velocity is 0 rad/s because the bike comes to a stop. The initial angular velocity is given as 17.7 rad/s, and you've already calculated the time. Plug in the values:
Angular acceleration = (0 rad/s - 17.7 rad/s) / time
Now you have all the information to calculate the answers. Plug in the values and do the calculation.