A wheel has a radius of 3.1 m. How far (path length) does a point on the circumference travel if the wheel is rotated through the following angles, respectively?
(a) 36°
m
(b) 36 rad
m
(c) 36 rev
m
circumference = 2 pi r
= 19.48 meters
a) 36/360 = .1
.1 * 19.48 = 1.948 meters
b) angle in radians * r = distance
36 * 3.1 = 111.6 meters
c) 36 * 19.48 = 701
To determine the distance traveled by a point on the circumference of a wheel, we need to calculate the path length. The path length is calculated by multiplying the circumference of the wheel by the fraction of the revolution completed.
First, we need to calculate the circumference of the wheel using the formula:
Circumference = 2 * π * radius
Given that the radius of the wheel is 3.1 m, we can calculate the circumference as follows:
Circumference = 2 * π * 3.1 = 19.42 m
(a) To calculate the distance traveled for an angle of 36°, we need to find the fraction of the revolution completed. Since one full revolution is equal to 360°, we can calculate the fraction as follows:
Fraction of revolution = (Angle in degrees) / 360° = 36° / 360° = 0.1
To find the distance traveled, we multiply the circumference by the fraction of the revolution:
Distance = Circumference * Fraction of revolution = 19.42 m * 0.1 = 1.942 m
Therefore, the point on the circumference will travel a path length of 1.942 meters when the wheel is rotated by 36°.
(b) To calculate the distance traveled for an angle of 36 radians, we need to find the fraction of the revolution completed. Since there are 2π radians in a full revolution, we can calculate the fraction as follows:
Fraction of revolution = Angle in radians / (2π) = 36 rad / (2π) ≈ 5.73
To find the distance traveled, we multiply the circumference by the fraction of the revolution:
Distance = Circumference * Fraction of revolution = 19.42 m * 5.73 ≈ 111.23 m
Therefore, the point on the circumference will travel a path length of approximately 111.23 meters when the wheel is rotated by 36 radians.
(c) To calculate the distance traveled for 36 revolutions, we can simply multiply the circumference by the number of revolutions:
Distance = Circumference * Number of revolutions = 19.42 m * 36 = 699.12 m
Therefore, the point on the circumference will travel a path length of 699.12 meters when the wheel is rotated by 36 revolutions.