radians

(noun)

The angle subtended at the centre of a circle by an arc of the circle of the same length as the circle's radius.

Examples of radians in the following topics:

  • Angular Position, Theta

    • This result is the basis for defining the units used to measure rotation angles to be radians (rad), defined so that:
    • Because there are 360º in a circle or one revolution, the relationship between radians and degrees is thus 2π rad=360º, so that:
    • Assess the relationship between radians the the revolution of a CD

  • Angular vs. Linear Quantities

    • The units of angular velocity are radians per second.
    • Radian describes the plane angle subtended by a circular arc as the length of the arc divided by the radius of the arc.
    • One radian is the angle subtended at the center of a circle by an arc that is equal in length to the radius of the circle.
    • More generally, the magnitude in radians of such a subtended angle is equal to the ratio of the arc length to the radius of the circle; that is, $\theta = \frac{s}{r}$, where $\theta$ is the subtended angle in radians, $s$ is arc length, and $r$ is radius.

  • Rotational Angle and Angular Velocity

    • The amount the object rotates is called the rotational angle and may be measured in either degrees or radians.
    • Since the rotational angle is related to the distance $\Delta S$ and to the radius $r$ by the equation $\Delta \theta = \frac{\Delta S}{R}$, it is usually more convenient to use radians.
    • This will give the angular velocity, usually denoted by $\omega$, in terms of radians per second.

  • Period and Frequency

    • Angular frequency is often represented in units of radians per second (recall there are 2π radians in a circle).
    • The locomotive's wheels spin at a frequency of f cycles per second, which can also be described as ω radians per second.

  • Simple Harmonic Motion and Uniform Circular Motion

    • For a path around a circle of radius r, when an angle θ (measured in radians) is swept out, the distance traveled on the edge of the circle is s = rθ.
    • You can prove this yourself by remembering that the circumference of a circle is 2*pi*r, so if the object traveled around the whole circle (one circumference) it will have gone through an angle of 2pi radians and traveled a distance of 2pi*r.
    • The angular velocity ω is in radians per unit time; in this case 2π radians is the time for one revolution T.

  • Problem-Solving Techniques

    • Be sure to use units of radians for angles.
    • Before using this equation, we must convert the number of revolutions into radians, because we are dealing with a relationship between linear and rotational quantities:

  • Centripetal Force

    • As the object travels its path, it sweeps out an arc that can be measured in degrees or radians.

  • Angular Acceleration, Alpha

    • In SI units, it is measured in radians per second squared (rad/s2), and is usually denoted by the Greek letter alpha ($\alpha$).

  • Position, Velocity, and Acceleration as a Function of Time

    • A general form of a sinusoidal wave is $y(x,t) = A sin(kx-\omega t + \phi)$, where A is the amplitude of the wave, $\omega$ is the wave's angular frequency, k is the wavenumber, and $\phi$ is the phase of the sine wave given in radians.
    • In this graph, the angle x is given in radians (π = 180°).

  • Phasors