A cyclist starts from rest and pedals so that the wheels make 8.1 revolutions in the first 5.4 s. What is the angular acceleration of the wheels (assumed constant)?
To find the angular acceleration of the wheels, we can use the formula:
angular acceleration (α) = (final angular velocity - initial angular velocity) / time
The initial angular velocity is 0 because the cyclist starts from rest.
To find the final angular velocity, we need to first find the total angle covered by the wheels.
Given that the wheels make 8.1 revolutions, we can calculate the total angle using the conversion:
total angle = 2π × number of revolutions
total angle = 2π × 8.1
Next, we calculate the final angular velocity:
final angular velocity (ω) = total angle / time
final angular velocity = (2π × 8.1) / 5.4
Now, we can substitute the values into the formula to find the angular acceleration:
angular acceleration (α) = (final angular velocity - initial angular velocity) / time
angular acceleration = (final angular velocity - 0) / 5.4
Simplifying this expression gives us the angular acceleration.
To find the angular acceleration of the wheels, we need to use the formula:
angular acceleration (α) = (final angular velocity (ω) - initial angular velocity (ω0)) / time (t)
We are given that the cyclist starts from rest, so the initial angular velocity (ω0) is 0. We are also given that the wheels make 8.1 revolutions in the first 5.4 seconds.
To find the final angular velocity (ω), we need to convert the number of revolutions into the corresponding angular displacement. Since one revolution is equal to 2π radians, the angular displacement can be calculated as:
angular displacement (θ) = 8.1 revolutions * 2π radians/revolution
Now, we can calculate the final angular velocity using the formula:
ω = θ / t
Substituting the values, we get:
ω = (8.1 revolutions * 2π radians/revolution) / 5.4 s
Once you calculate the final angular velocity (ω), you can substitute it along with the initial angular velocity (ω0 = 0) and the given time (t = 5.4 s) into the formula for angular acceleration (α) to get the answer.