Explain how 1 radian is equal to 1/2π revolutions.
The Correct Answer and Explanation is:
To understand how 1 radian is equal to ( \frac{1}{2\pi} ) revolutions, we first need to define both radians and revolutions in the context of circular motion.
Definitions:
- Radian: A radian is defined as the angle subtended at the center of a circle by an arc whose length is equal to the radius of the circle. Therefore, if you have a circle of radius ( r ), an angle of 1 radian corresponds to an arc length of ( r ).
- Revolution: A full revolution around a circle refers to a complete rotation of 360 degrees, which is equivalent to ( 2\pi ) radians. Thus, one revolution corresponds to an angle of ( 2\pi ) radians.
Derivation:
To find the relationship between radians and revolutions, we can set up the following equivalence based on the definitions:
- Since one revolution corresponds to ( 2\pi ) radians, we can say:
[
1 \text{ revolution} = 2\pi \text{ radians}
]
To express 1 radian in terms of revolutions, we can rearrange the equation:
[
1 \text{ radian} = \frac{1}{2\pi} \text{ revolutions}
]
Explanation:
This relationship emerges from the definition of the circle itself. The circumference of a circle, which is the total distance around the circle, is given by the formula ( C = 2\pi r ), where ( r ) is the radius. When we consider the unit circle (a circle with a radius of 1), the circumference becomes ( 2\pi ).
When the arc length equals the radius (1 unit), we have defined 1 radian. Since the full circumference of the unit circle represents one complete revolution (or ( 2\pi ) radians), we conclude that the angle of 1 radian is simply the fraction of the total circumference represented by that arc length. Therefore, dividing the total number of radians in a revolution by ( 2\pi ) gives us the fraction of a revolution that corresponds to 1 radian, leading us to the conclusion that:
[
1 \text{ radian} = \frac{1}{2\pi} \text{ revolutions}
]
This mathematical relationship illustrates the connection between linear distance traveled along a circle and the angular measurement in radians and revolutions.