To find the answers to these questions, we can use the formula relating linear speed and angular speed.
Linear speed (v) = angular speed (Ļ) Ć radius (r)
The formula for the circumference of a circle is given by:
Circumference = 2Ļ Ć radius
(a) To find the angular speed of the tires about their axles, we can convert the linear speed from km/h to m/s. Then we can use the formula:
Angular speed = Linear speed / Circumference
First, let's convert the tire diameter from centimeters to meters:
Diameter = 100.0 cm = 1 meter (since 1 cm = 0.01 m)
The radius of the tire is half the diameter, so the radius (r) = 1/2 Ć 1 = 0.5 meters.
Now, let's convert the linear speed from km/h to m/s:
Speed = 50.0 km/h = 50.0 Ć (1000/3600) m/s = 13.89 m/s (rounded to two decimal places)
Since the tire's distance traveled in one revolution is equal to the circumference of the tire, which is 2Ļ Ć radius, the angular speed can be calculated as follows:
Angular speed = 13.89 m/s / (2Ļ Ć 0.5 m)
= 13.89 m/s / 3.14 m
ā 4.42 rad/s (rounded to two decimal places)
Therefore, the angular speed of the tires about their axles is approximately 4.42 rad/s.
(b) To find the magnitude of the angular acceleration of the wheels, we can use the formula:
Angular acceleration = (Final angular speed - Initial angular speed) / Time
Since the car was brought to a stop uniformly in 25.0 complete turns of the tires, the final angular speed is 0 rad/s (since it stopped).
The initial angular speed can be calculated as follows:
Initial angular speed = Angular speed Ć 2Ļ (since one complete turn is equal to 2Ļ radians)
Initial angular speed = 4.42 rad/s Ć 2Ļ ā 27.77 rad/s (rounded to two decimal places)
Now, let's calculate the angular acceleration:
Angular acceleration = (0 rad/s - 27.77 rad/s) / 25.0 turns
Angular acceleration = -27.77 rad/s / 25.0 turns
ā -1.11 rad/sĀ² (rounded to two decimal places)
Therefore, the magnitude of the angular acceleration of the wheels is approximately 1.11 rad/sĀ².
(c) To calculate how far the car moves during the braking, we need to find the distance traveled by one tire in 25.0 complete turns.
Distance traveled by one tire = Circumference Ć Number of turns
Circumference = 2Ļ Ć radius = 2Ļ Ć 0.5 m = Ļ m (since Ļ ā 3.14)
Distance traveled by one tire = Ļ m Ć 25.0 turns
= 25Ļ m
Since there are two tires on the car, the total distance moved by the car during the braking is:
Total distance moved by the car = 2 Ć Distance traveled by one tire
= 2 Ć 25Ļ m
ā 157.08 m (rounded to two decimal places)
Therefore, the car moves approximately 157.08 meters during the braking.