In Polar Coordinates
During circular motion the body moves on a curve that can be described in polar coordinate system as a fixed distance R from the center of the orbit taken as origin, oriented at an angle θ (t) from some reference direction. See Figure 2. The displacement vector is the radial vector from the origin to the particle location:
where is the unit vector parallel to the radius vector at time t and pointing away from the origin. It is convenient to introduce the unit vector orthogonal to as well, namely . It is customary to orient to point in the direction of travel along the orbit.
The velocity is the time derivative of the displacement:
Because the radius of the circle is constant, the radial component of the velocity is zero. The unit vector has a time-invariant magnitude of unity, so as time varies its tip always lies on a circle of unit radius, with an angle θ the same as the angle of . If the particle displacement rotates through an angle dθ in time dt, so does, describing an arc on the unit circle of magnitude dθ. See the unit circle at the left of Figure 2. Hence:
where the direction of the change must be perpendicular to (or, in other words, along ) because any change d in the direction of would change the size of . The sign is positive, because an increase in dθ implies the object and have moved in the direction of . Hence the velocity becomes:
The acceleration of the body can also be broken into radial and tangential components. The acceleration is the time derivative of the velocity:
The time derivative of is found the same way as for . Again, is a unit vector and its tip traces a unit circle with an angle that is π/2 + θ. Hence, an increase in angle dθ by implies traces an arc of magnitude dθ, and as is orthogonal to, we have:
where a negative sign is necessary to keep orthogonal to . (Otherwise, the angle between and would decrease with increase in dθ.) See the unit circle at the left of Figure 2. Consequently the acceleration is:
The centripetal acceleration is the radial component, which is directed radially inward:
while the tangential component changes the magnitude of the velocity:
Other articles related to "polar, coordinates, polar coordinate, coordinate":
... Polar aprotic solvents are solvents that will dissolve many salts, but lack an acidic hydrogen ... Although discouraging use of the term "polar aprotic", IUPAC describes such solvents as having both high dielectric constants and high dipole moments, an example being ... fail this criterion) solvents dissolve organic salts such as tetraethylammonium iodide Polar aprotic solvents are often essential for reactions that involve strong bases, such as ...
... In geometry, the polar angle may be one of the two coordinates of a two-dimensional polar coordinate system one of the three coordinates of a three-dimensional spherical coordinate system in this ...
... The Polar Circle Marathon is a classical 42.195 km marathon, which however has the very untraditional feature that it is situated on 66 degrees northern ... The Polar Circle Marathon is more demanding than usual marathons due to the cold weather and the slippery surface for the parts of the race that goes over the actual ice cap ... The Polar Circle Marathon was held for the 5th time in history on 18 October 2008 around Kangerlussuaq in Western Greenland ...
... where he served as property and disbursing officer before becoming responsible for planning all polar expeditions ... Under his auspices, he was responsible for polar expeditions at a time when several countries were also supporting polar exploration and scientific studies ...
... There were 354 households out of which 38.4% had children under the age of 18 living with them, 68.1% were married couples living together, 6.5% had a female householder with no husband present, and 20.6% were non-families. 17.8% of all households were made up of individuals and 7.3% had someone living alone who was 65 years of age or older ...
Famous quotes containing the word polar:
“Professor Fate: My apologies. Theres a polar bear in our car.”
—Arthur Ross. Professor Fate (Jack Lemmon)