You are holding the axle of a bicycle wheel with radius 30 cm and mass 1.07 kg. You get the wheel spinning at a rate of 79 rpm and then stop it by pressing the tire against the pavement. You notice that it takes 1.11 s for the wheel to come to a complete stop. What is the angular acceleration of the wheel?
To find the angular acceleration of the wheel, we can use the equation:
angular acceleration = (angular velocity final - angular velocity initial) / time
First, we need to convert the initial angular velocity from rpm to rad/s. We know that 1 revolution is equal to 2π radians. So the initial angular velocity in rad/s can be calculated as follows:
initial angular velocity = (79 rpm) * (2π rad/1 min) * (1 min/60 s)
Now, we need to convert the time from seconds to minutes, as the angular velocity is measured in rad/min:
time in minutes = 1.11 s * (1 min/60 s)
Finally, we can substitute the values into the equation to find the angular acceleration:
angular acceleration = (0 rad/min - initial angular velocity) / time in minutes
Let's calculate the angular acceleration step by step:
1. Convert the initial angular velocity from rpm to rad/s:
initial angular velocity = (79 rpm) * (2π rad/1 min) * (1 min/60 s)
≈ 8.27 rad/s
2. Convert the time from seconds to minutes:
time in minutes = 1.11 s * (1 min/60 s)
≈ 0.0185 min
3. Calculate the angular acceleration:
angular acceleration = (0 rad/min - 8.27 rad/s) / 0.0185 min
≈ -445.95 rad/min^2
The angular acceleration of the wheel is approximately -445.95 rad/min^2. The negative sign indicates that the wheel is decelerating (slowing down) in its rotation.
To find the angular acceleration of the wheel, we can use the formula for angular acceleration:
Angular acceleration (α) = (final angular velocity - initial angular velocity) / time
First, we need to convert the initial angular velocity from rpm to rad/s. Since 1 revolution is equal to 2π radians, we can convert rpm to rad/s using the following conversion factor:
1 rpm = 2π/60 rad/s
Now, let's calculate the initial angular velocity:
Initial angular velocity (ω_i) = 79 rpm * (2π/60 rad/s) = 8.303 rad/s
Next, we can calculate the final angular velocity. The wheel comes to a complete stop, which means the final angular velocity is 0:
Final angular velocity (ω_f) = 0 rad/s
Now, we can substitute the values into the formula for angular acceleration:
Angular acceleration (α) = (0 - 8.303 rad/s) / 1.11 s
Let's calculate the angular acceleration:
Angular acceleration (α) = -7.48 rad/s^2
Therefore, the angular acceleration of the wheel is approximately -7.48 rad/s^2. The negative sign indicates that the wheel is decelerating (slowing down).