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Concept
Angular displacement
Summary
The angular displacement (symbol θ, , or φ), also called angle of rotation or rotational displacement, of a physical body is the angle (in units of radians, degrees, turns, etc.) through which the body rotates (revolves or spins) around a centre or axis of rotation. Angular displacement may be signed, indicating the sense of rotation (e.g., clockwise); it may also be greater (in absolute value) than a full turn.
When a body rotates about its axis, the motion cannot simply be analyzed as a particle, as in circular motion it undergoes a changing velocity and acceleration at any time. When dealing with the rotation of a body, it becomes simpler to consider the body itself rigid. A body is generally considered rigid when the separations between all the particles remains constant throughout the body's motion, so for example parts of its mass are not flying off. In a realistic sense, all things can be deformable, however this impact is minimal and negligible.
In the example illustrated to the right (or above in some mobile versions), a particle or body P is at a fixed distance r from the origin, O, rotating counterclockwise. It becomes important to then represent the position of particle P in terms of its polar coordinates (r, θ). In this particular example, the value of θ is changing, while the value of the radius remains the same. (In rectangular coordinates (x, y) both x and y vary with time). As the particle moves along the circle, it travels an arc length s, which becomes related to the angular position through the relationship:
Angular displacement may be expressed in radians or degrees. Using radians provides a very simple relationship between distance traveled around the circle (circular arc length) and the distance r from the centre (radius):
For example, if a body rotates 360° around a circle of radius r, the angular displacement is given by the distance traveled around the circumference - which is 2πr - divided by the radius: which easily simplifies to: . Therefore, 1 revolution is radians.
Official source
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