15. erik rides his bike 4.5 km every morning.\na) if the radius of the tire on his bike is 32 cm, determine…

15. erik rides his bike 4.5 km every morning.\na) if the radius of the tire on his bike is 32 cm, determine the number of radians that the tire will rotate during the entire trip.\nb) find the angular velocity in radians per second if the tire turns at 150 rev/min.\nangular velocity = \\frac{total degrees travelled}{time}\n4.1 radian measure • mhr 79

15. erik rides his bike 4.5 km every morning.\na) if the radius of the tire on his bike is 32 cm, determine the number of radians that the tire will rotate during the entire trip.\nb) find the angular velocity in radians per second if the tire turns at 150 rev/min.\nangular velocity = \\frac{total degrees travelled}{time}\n4.1 radian measure • mhr 79

Answer

Explanation:

Step1: Convert the distance to centimeters

Since (1\ km = 100000\ cm), then (4.5\ km=4.5\times100000 = 450000\ cm).

Step2: Find the circumference of the tire

The formula for the circumference of a circle is (C = 2\pi r), where (r = 32\ cm). So (C=2\pi\times32=64\pi\ cm).

Step3: Calculate the number of rotations

The number of rotations (n=\frac{\text{Total distance}}{\text{Circumference}}=\frac{450000}{64\pi}).

Step4: Find the number of radians

One full - rotation is (2\pi) radians. So the number of radians (\theta=n\times2\pi=\frac{450000}{64\pi}\times2\pi=\frac{450000}{32}=14062.5) radians.

Step5: Convert revolutions per minute to radians per second for part (b)

We know that (1) revolution ( = 2\pi) radians and (1) minute (=60) seconds. If the tire turns at (150\ rev/min), then the angular velocity (\omega) in radians per second: First, for (150) revolutions, the number of radians is (150\times2\pi) radians in (1) minute ((60) seconds). (\omega=\frac{150\times2\pi}{60}=5\pi\approx 15.71) radians per second.

Answer:

a) (14062.5) radians b) (5\pi\approx15.71) radians per second