Final Answer:
a. The tire's angular velocity is approximately 318.31 rpm.
b. The speed of a point at the top edge of the tire is about 31.42 m/s.
Step-by-step explanation
For part a, the angular velocity of the tire can be found using the formula: angular velocity (in radians/second) = linear velocity / radius. First, convert the speed from meters per second to centimeters per second (20 m/s = 2000 cm/s) for consistency with the diameter in centimeters (60 cm). The circumference of the tire is π times its diameter, so the radius is half the diameter (30 cm). Using the formula, angular velocity = (2000 cm/s) / (30 cm) = approximately 66.67 radians/second. Convert this value to revolutions per minute (rpm) by multiplying by 60 seconds/minute and dividing by 2π radians/revolution: (66.67 * 60) / (2π) ≈ 318.31 rpm.
For part b, the speed at the top edge of the tire (assuming no slipping) equals the sum of the linear speed due to the car's forward motion and the tangential speed due to rotation. The linear speed remains 20 m/s, and the tangential speed at the edge can be calculated using the formula for the circumference of a circle (2πr). Therefore, the total speed = linear speed + tangential speed = 20 m/s + (2π * 30 cm * 66.67 rad/s) = 20 m/s + approximately 628.32 cm/s ≈ 31.42 m/s.
This results in an angular velocity of approximately 318.31 rpm for the tire and a speed of about 31.42 m/s at the top edge.
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