Angular displacement

Angular displacement

Angular displacement of a body is the angle in radians (degrees, revolutions) through which a point or line has been rotated in a specified sense about a specified axis.

When an object rotates about its axis, the motion cannot simply be analyzed as a particle, since in circular motion it undergoes a changing velocity and acceleration at any time (t). When dealing with the rotation of an object, 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 objects 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. Thus the rotation of a rigid body over a fixed axis is referred to as rotational motion.

In the example illustrated to the right, a particle on object P 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, theta). In this particular example, the value of theta is changing, while the value of the radius remains the same. (In rectangular coordinates (x, y) both x and y are going to 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:

s=rtheta ,

Angular Displacement is measured in radians rather than degrees. This is because it provides a very simple relationship between distance traveled around the circle and the distance r from the centre.

theta=frac sr

For example if an object rotates 360 degrees around a circle radius r the angular displacement is given by the distance traveled the circumference which is 2Pi r Divided by the radius in: theta= frac{2pi r}r which easily simplifies to 2pi. Therefore 1 revolution is 2pi radians.

When object travels from point P to point Q, as it does in the illustration to the left, over delta t the radius of the circle goes around a change in angle. Delta theta = Delta theta_2 - Delta theta_1 which equals the Angular Displacement.

In three dimensions, angular displacement has a direction and a magnitude. The direction specifies the axis of rotation; the magnitude specifies the rotation in radians about that axis (using the right-hand rule to determine direction). Despite having direction and magnitude, angular displacement is not a vector because it does not obey the commutative law.

Angular Velocity

As with linear motion, we define the average angular speed (omega) as the ratio of this angular displacement to the time interval Delta t:

bar{boldsymbolomega}={Delta theta over Delta t}

Thus the instantaneous angular velocity can be retained by an infinitely small change in time, which is simply finding the derivative of angular displacement with respect to time.:

boldsymbolomega ={mathrm{d}theta over mathrm{d}t} = lim_{Delta t to 0}{Delta theta over Delta t}

Angular speed is measured in units of radians per second, or s^{-1}, since radians carry no unit and are dimensionless. The same work can then be done to find the value for angular acceleration.

In modern application, mostly all scientific reality is built on the concepts of angular displacement. It can be said that all measurements of physical properties are quantized in terms of the angular displacement of some reference system. Time is the measure of the reference angular displacement between two events associated with one body, space is the measure of the reference angular displacement between two events associated with two different bodies, mass is a function of time and space (Kepler's Law), and all other physical properties are quantized in terms of these three properties (time, space and mass). The whole idea comes from this concept of only knowing the value of certain things, in relation to something else, because without these other quantities, values become meaningless.

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