A tire placed on a balancing machine in a service station starts from rest and turns through 4.82 revolutions in 1.48 s before reaching its final angular speed. Calculate its angular acceleration.
To calculate the angular acceleration of the tire, we can use the following formula:
Angular acceleration (α) = (final angular speed - initial angular speed) / time
The given information in the problem is as follows:
- Initial angular speed (ω initial) = 0 rad/s (since the tire starts from rest)
- Final angular speed (ω final) = unknown (to be determined)
- Time (t) = 1.48 s
- Number of revolutions (N) = 4.82
To find the final angular speed, we first need to convert the number of revolutions to radians.
1 revolution is equal to 2π radians.
Therefore, the total angle covered by the tire is:
θ = 2π * N
Now, we can determine the final angular speed using the formula:
ω final = θ / t
Substituting the values:
ω final = (2π * N) / t
Finally, we can calculate the angular acceleration using the formula mentioned above:
α = (ω final - ω initial) / t
Now, let's plug in the values and solve the equation step by step:
First, convert the number of revolutions to radians:
θ = 2π * N = 2π * 4.82 = 30.26 radians
Then, calculate the final angular speed:
ω final = θ / t = 30.26 / 1.48 ≈ 20.44 rad/s
Finally, find the angular acceleration:
α = (ω final - ω initial) / t = (20.44 - 0) / 1.48 ≈ 13.84 rad/s^2
Therefore, the angular acceleration of the tire is approximately 13.84 rad/s^2.