In physics, uniform circular motion describes the motion of a body traversing a circular path at constant speed. The distance of the body from the axis of rotation remains constant at all times. Though the body’s speed is constant, its velocity is not constant: velocity, a vector quantity, depends on both the body’s speed and its direction of travel. This changing velocity indicates the presence of an acceleration; this centripetal acceleration is of constant magnitude and directed at all times towards the axis of rotation. This acceleration is, in turn, produced by a centripetal force which is also constant in magnitude and directed towards the axis of rotation.
In the case of rotation around a fixed axis of a rigid body that is not negligibly small compared to the radius of the path, each particle of the body describes a uniform circular motion with the same angular velocity, but with velocity and acceleration varying with the position with respect to the axis.
 Formulas for uniform circular motion
For motion in a circle of radius r, the circumference of the circle is C = 2π r. If the period for one rotation is T, the angular rate of rotation, also known as angular velocity, ω is:
- and the units are radians/sec
The speed of the object traveling the circle is:
The angle θ swept out in a time t is:
The acceleration due to change in the direction is:
The vector relationships are shown in Figure 1. The axis of rotation is shown as a vector Ω perpendicular to the plane of the orbit and with a magnitude ω = dθ / dt. The direction of Ω is chosen using the right-hand rule. With this convention for depicting rotation, the velocity is given by a vector cross product as
which is a vector perpendicular to both Ω and r ( t ), tangential to the orbit, and of magnitude ω r. Likewise, the acceleration is given by
which is a vector perpendicular to both Ω and v ( t ) of magnitude ω |v| = ω2 r and directed exactly opposite to r ( t ).
In the simplest case the speed, mass and radius are constant.
Consider a body of one kilogram, moving in a circle of radius one metre, with an angular velocity of one radian per second.
- The speed is one metre per second.
- The inward acceleration is one metre per square second[v^2/r]
- It is subject to a centripetal force of one kilogram metre per square second, which is one newton.
- The momentum of the body is one kg·m·s−1.
- The moment of inertia is one kg·m2.
- The angular momentum is one kg·m2·s−1.
- The kinetic energy is 1/2 joule.
- The circumference of the orbit is 2π (~ 6.283) metres.