Circular Motion - Uniform Circular Motion - in Polar Coordinates

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:

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

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