Final answer:
The angular speed in radians per second (rad/s) can be estimated by calculating the initial and final angular speeds given the linear speeds and tire dimensions, and then taking the average of these as the speed is reduced uniformly. For the car described, the initial angular speed is 59.34 rad/s, the final angular speed is 40.41 rad/s, and the estimated average is 49.88 rad/s.
Step-by-step explanation:
The question involves finding the angular speed in radians per second (rad/s) of a car tire as the speed of the car decreases uniformly from 94.0 km/h to 64.0 km/h, given that the tire has made 88 revolutions and has a diameter of 0.88 m. To solve this, we need to convert the linear speeds to angular speeds and utilize the relationship between the linear distance traveled during the revolutions and the circumference of the tires.
First, convert the car's speeds from km/h to m/s:
- 94.0 km/h = (94.0 * 1000 m) / (3600 s) = 26.11 m/s
- 64.0 km/h = (64.0 * 1000 m) / (3600 s) = 17.78 m/s
Now, calculate the angular speed using the formula v = rω where v is the linear speed, r is the radius of the tire, and ω is the angular speed.
The radius (r) is half of the diameter, so r = 0.88 m / 2 = 0.44 m.
The initial angular speed (ωi) when the speed is 94.0 km/h (26.11 m/s):
ωi = v / r = 26.11 m/s / 0.44 m = 59.34 rad/s
The final angular speed (ωf) when the speed is 64.0 km/h (17.78 m/s):
ωf = v / r = 17.78 m/s / 0.44 m = 40.41 rad/s
We can use the average of the initial and final angular speeds to estimate the angular speed during the deceleration since the problem states it changes uniformly. So the average angular speed (ωavg)= (ωi + ωf) / 2 = (59.34 rad/s + 40.41 rad/s) / 2 = 49.88 rad/s. This is the estimated angular speed in radians per second.