The tires of a car make 63 revolutions as the car reduces its speed uniformly from 94.0 km/h to 58.0 km/h. The tires have a diameter of 0.88 m. What was the angular acceleration of each tire? And If the car continues to decelerate at this rate, how much more time is required for it to stop?

To find the angular acceleration of each tire, we first need to calculate the change in angular velocity (∆ω) and the change in time (∆t).

1. Convert the initial and final speeds from km/h to m/s:
Initial speed (v₁) = 94.0 km/h = 94.0 * (1000/3600) m/s = 26.1 m/s
Final speed (v₂) = 58.0 km/h = 58.0 * (1000/3600) m/s = 16.1 m/s

2. Calculate the change in angular velocity (∆ω):
∆ω = (v₂ - v₁) / r
= (16.1 - 26.1) / (0.88 / 2)
= -10.0 / 0.44
= -22.7 rad/s

(Note: Since the car is decelerating, the angular velocity change is negative.)

3. Calculate the change in time (∆t):
We know that the number of revolutions (∆θ) is equal to the number of complete circles the tire makes. The relationship between angular displacement (∆θ) and the number of revolutions is given by:
∆θ = 2π * N
where N is the number of revolutions.

Given that the number of revolutions (∆θ) is 63, we can calculate the change in time (∆t) using the formula:
∆t = ∆θ / ∆ω
= 63 / (-22.7)
≈ -2.77 s

(Note: Since the car is decelerating, the change in time (∆t) is negative.)

Therefore, the angular acceleration (∝) of each tire is given by:
∝ = ∆ω / ∆t
= (-22.7) / (-2.77)
≈ 8.19 rad/s²

If the car continues to decelerate at this rate, we can calculate the time required for it to stop.

4. Calculate the time required for the car to stop (t):
We can use the equation of motion:

v = v₀ - ∝t

where v₀ is the initial velocity (26.1 m/s), v is the final velocity (0 m/s), ∝ is the angular acceleration (-8.19 rad/s²), and t is the time.

Substituting the values, we get:

0 = 26.1 - (-8.19)t
8.19t = 26.1
t ≈ 3.19 s

Therefore, if the car continues to decelerate at the same rate, it will take approximately 3.19 seconds for it to stop completely.

wf^2=wi^2+2*angularacceleration*63*2PI

to get wf and wi, speed/(2PI*d) = w

change speed to m/s