A tire placed on a balancing machine in a service station starts from rest and turns through 4.71 revolutions in 1.29 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 velocity - initial angular velocity) / time
Here, the initial angular velocity is 0 since the tire starts from rest. The final angular velocity can be calculated by dividing the number of revolutions by the time:
Final angular velocity (ω) = (4.71 revolutions) / (1.29 s)
Now, let's substitute these values into the formula to find the angular acceleration:
Angular acceleration (α) = [(4.71 revolutions / 1.29 s) - 0] / 1.29 s
Simplifying the equation:
Angular acceleration (α) = (4.71 revolutions / 1.29 s^2)
Therefore, the angular acceleration of the tire is 3.65 revolutions/second^2.
To calculate the angular acceleration, we need to use the equation:
Angular acceleration (α) = Change in Angular Velocity (Δω) / Time taken (Δt)
The given information is:
Number of revolutions, n = 4.71 revolutions
Time, t = 1.29 s
First, we need to convert the number of revolutions into radians:
1 revolution = 2π radians
So, the number of radians in 4.71 revolutions is:
θ = 4.71 revolutions * 2π radians/revolution
Next, we calculate the change in angular velocity Δω:
The initial angular velocity, ω_initial, is 0 (since the tire starts from rest).
The final angular velocity, ω_final, can be calculated using the formula:
ω_final = θ / t
Finally, we can calculate the angular acceleration (α) using the formula mentioned earlier:
α = Δω / Δt
Let's calculate the angular acceleration step by step:
Step 1: Convert the number of revolutions into radians:
θ = 4.71 revolutions * 2π radians/revolution
θ = 4.71 * 2π radians
Step 2: Calculate the final angular velocity (ω_final):
ω_final = θ / t
ω_final = (4.71 * 2π) / 1.29
Step 3: Calculate the change in angular velocity (Δω):
Δω = ω_final - ω_initial
Since the initial angular velocity (ω_initial) is 0, Δω will be equal to ω_final.
Step 4: Calculate the angular acceleration (α):
α = Δω / Δt
α = ω_final / t
Now, let's substitute the values and calculate the angular acceleration.