A wheel has a radius of 3.5 m. How far (path length) does a point on the circumference travel if the wheel is rotated through an angle of 71.0°? B) A wheel has a radius of 3.5 m. How far (path length) does a point on the circumference travel if the wheel is rotated through an angle of 71.0 rad? A wheel has a radius of 3.5 m. How far (path length) does a point on the circumference travel if the wheel is rotated through an angle of 71.0 revolutions?
pathlength=radius*angleinRadians.
a)2π . 3.5 . 71/ 360 = 4.33m
b) 2π . 3.5 . 71/ 2π = 248.5m
c) 2π . 3.5 . 71= 1561.37m
To find the path length, or the distance traveled by a point on the circumference of a wheel, we need to use the formula:
Path Length = Circumference of the wheel x (angle in radians or revolutions)
In this case, we are given the radius of the wheel, which is 3.5 m.
A) If the wheel is rotated through an angle of 71.0°, we need to convert this angle to radians before using the formula.
To convert degrees to radians, we multiply the angle in degrees by π/180.
Angle in radians = 71.0° x π/180 ≈ 1.239 radians
Now, we use the formula:
Path Length = Circumference of the wheel x Angle in radians
The circumference of a circle can be calculated using the formula: Circumference = 2πr, where r is the radius.
Circumference of the wheel = 2π(3.5) = 2π(3.5) = 7π m (approximately 21.99 m)
Path Length = 7π m x 1.239 radians ≈ 8.635 m
Therefore, the point on the circumference of the wheel will travel approximately 8.635 meters if the wheel is rotated through an angle of 71.0°.
B) If the wheel is rotated through an angle of 71.0 rad, we can directly use the formula:
Path Length = Circumference of the wheel x Angle in radians
Using the same circumference of the wheel (7π m) and the given angle (71.0 rad), we can calculate:
Path Length = 7π m x 71.0 rad ≈ 497.35 m
Therefore, the point on the circumference of the wheel will travel approximately 497.35 meters if the wheel is rotated through an angle of 71.0 rad.
C) If the wheel is rotated through an angle of 71.0 revolutions, we need to convert this angle to radians.
To convert revolutions to radians, we multiply the angle in revolutions by 2π.
Angle in radians = 71.0 revolutions x 2π ≈ 447.09 radians
Now, we use the formula:
Path Length = Circumference of the wheel x Angle in radians
Using the same circumference of the wheel (7π m) and the given angle (447.09 radians), we can calculate:
Path Length = 7π m x 447.09 radians ≈ 3139.63 m
Therefore, the point on the circumference of the wheel will travel approximately 3139.63 meters if the wheel is rotated through an angle of 71.0 revolutions.