An automobile traveling at 50.0 km/h has tires of 100.0 cm diameter.

(a) What is the angular speed of the tires about their axles?
1 rad/s

(b) If the car is brought to a stop uniformly in 25.0 complete turns of the tires (without skidding), what is the magnitude of the angular acceleration of the wheels?
2 rad/s2

(c) How far does the car move during the braking?
3 m

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Remember
Torque = rate of change of angular momentum
= I alpha

omega = w = angular speed
angular momentum = I w

alpha = dw/dt = rate of change of w = angular acceleration

tangential speed = v = w r

distance rolled = r theta = 2 pi r*number of turns

To find the answers to these questions, we can use the formula relating linear speed and angular speed.

Linear speed (v) = angular speed (ω) × radius (r)

The formula for the circumference of a circle is given by:

Circumference = 2π × 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 × 1 = 0.5 meters.

Now, let's convert the linear speed from km/h to m/s:

Speed = 50.0 km/h = 50.0 × (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 2π × radius, the angular speed can be calculated as follows:

Angular speed = 13.89 m/s / (2π × 0.5 m)
= 13.89 m/s / 3.14 m
≈ 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 × 2π (since one complete turn is equal to 2π radians)

Initial angular speed = 4.42 rad/s × 2π ≈ 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
≈ -1.11 rad/s² (rounded to two decimal places)

Therefore, the magnitude of the angular acceleration of the wheels is approximately 1.11 rad/s².

(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 × Number of turns

Circumference = 2π × radius = 2π × 0.5 m = π m (since π ≈ 3.14)

Distance traveled by one tire = π m × 25.0 turns
= 25π 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 × Distance traveled by one tire
= 2 × 25π m
≈ 157.08 m (rounded to two decimal places)

Therefore, the car moves approximately 157.08 meters during the braking.