The tires of a car make 58 revolutions as the car reduces its speed uniformly from 92 to 33 . The tires have a diameter of 0.86 .
What was the angular acceleration of the tires?
If the car continues to decelerate at this rate, how much more time is required for it to stop?
1.C = pi*D = 3.14*0.86 = 2.7Ft.
d = 58rev * 2.7Ft/rev = 156.6Ft.
Vo = 92mi/h * 5280Ft/mi *(1h/3600s)=134.9Ft/s. = Inital velocity.
Vf = 33mi/h * 5280Ft/mi * (1h/3600s)=48.4Ft/s. = Final velocity.
To = d/Vo = 156.6Ft / 134.9Ft/s = 1.16s.
Tf = 156.6Ft / 48.4Ft/s = 3.23s.
a = (Vf-Vo)/(Tf-To)
a = (48.4-134.9) / (3.23-1.16) = -41.8Ft/s^2, Linear.
a=-41.8Ft/s^2 * 6.28rad/2.7Ft=
-97.2rad/s^2, Angular.
2. Vf = Vo + at = 0.
48.4 -97.2t = 0,
t = 0.50s.
To find the angular acceleration of the tires, we can use the formula:
Angular acceleration (α) = (Final angular velocity (ω_f) - Initial angular velocity (ω_i)) / Time taken (t)
Given:
Number of revolutions = 58
Diameter of the tires = 0.86
First, let's find the initial and final angular velocities:
The initial angular velocity can be calculated using the formula:
ω_i = (2π * Initial speed) / (2π * Radius) = Initial speed / Radius
The final angular velocity can be calculated using the formula:
ω_f = (2π * Final speed) / (2π * Radius) = Final speed / Radius
Now, let's calculate the initial and final angular velocities:
Initial speed = 92
Final speed = 33
Radius = Diameter / 2 = 0.86 / 2 = 0.43
Using the above values:
Initial angular velocity (ω_i) = 92 / 0.43
Final angular velocity (ω_f) = 33 / 0.43
Now, we can calculate the angular acceleration:
Angular acceleration (α) = (ω_f - ω_i) / Time taken
To find the time taken, we can use the formula for the number of revolutions:
Number of revolutions = (ω_f - ω_i) * Time taken / (2π)
Let's rearrange the formula to solve for time taken:
Time taken = (Number of revolutions * 2π) / (ω_f - ω_i)
Now, we can substitute the values and solve for both the angular acceleration and the time taken.
To find the angular acceleration of the tires, we can use the formula:
Angular acceleration (α) = (Final angular velocity (ωf) - Initial angular velocity (ωi)) / Time (t)
Given that the car reduces its speed uniformly, we can assume that the initial angular velocity is equal to the final angular velocity. Therefore, ωi = ωf.
We are given that the car reduces its speed from 92 to 33. The angular velocity (ω) is related to the linear speed (v) and the radius (r) by the formula:
ω = v / r
We know the diameter (d) of the tires, so we can find the radius (r) by dividing the diameter by 2:
r = d / 2
Let's substitute the values into the formula to find the angular velocity:
ωi = vi / ri = 92 / (0.86 / 2)
ωf = vf / rf = 33 / (0.86 / 2)
Since ωi = ωf, we can find the common angular velocity (ω) using either ωi or ωf.
Let's use ωi = ωf to find ω:
ω = ωi = ωf = 92 / (0.86 / 2)
Now, to find the angular acceleration (α), we need the time (t). The number of revolutions made by the tires is given as 58. The number of revolutions (N) is related to the angular displacement (θ) by the formula:
N = θ / (2π)
We can find the angular displacement using the formula:
θ = ωi * t + (1/2) * α * t^2
We are given N = 58. Let's substitute the values into the formula to find the time (t):
58 = ωi * t + (1/2) * α * t^2
Since we want to find the angular acceleration (α), we need to rearrange the equation.
Let's multiply the equation by 2:
116 = 2 * (ωi * t + (1/2) * α * t^2)
Simplifying the equation:
116 = 2 * ωi * t + α * t^2
Now, let's substitute the value of ωi and solve the equation for α:
116 = 2 * (92 / (0.86 / 2)) * t + α * t^2
Now we can solve this equation to find the value of α.