**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 *d*θ in time *dt*, so does, describing an arc on the unit circle of magnitude *d*θ. 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 *d*θ 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 *d*θ by implies traces an arc of magnitude *d*θ, 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 *d*θ.) 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:

Read more about this topic: Circular Motion, Uniform Circular Motion

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