In
physics,
circular motion is
rotation along a
circle: a circular path or a circular
orbit. It can be
uniform, that is, with constant angular rate of rotation, or
non-uniform, that is, with a changing rate of rotation. The
rotation around a fixed axis of a three-dimensional body involves circular motion of its parts. The equations describing circular motion of an object do not take size or geometry into account, rather, the motion of a point mass in a plane is assumed. In practice, the
center of mass of a body can be considered to undergo circular motion.
Examples of circular motion include: an artificial satellite orbiting the Earth in
geosynchronous orbit, a stone which is tied to a rope and is being swung in circles (cf.
hammer throw), a racecar turning through a curve in a
race track, an electron moving perpendicular to a uniform
magnetic field, and a
gear turning inside a mechanism.
Circular motion is accelerated even if the angular rate of rotation is constant, because the object's velocity vector is constantly changing direction. Such change in direction of velocity involves
acceleration of the moving object by a
centripetal force, which pulls the moving object toward the center of the circular orbit. Without this acceleration, the object would move in a straight line, according to
Newton's laws of motion.
[edit] Formulas for uniform circular motion
Figure 1: Vector relationships for uniform circular motion; vector Ω representing the rotation is normal to the plane of the orbit. 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:

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| = ω
2 r and directed exactly opposite to
r (
t ).
[1]
[edit] Constant speed
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.
Then consider a body of
mass m, moving in a circle of radius
r, with an
angular velocity of
ω.
- The speed is v = r·ω.
- The centripetal (inward) acceleration is a = r·ω 2 = r −1·v 2.
- The centripetal force is F = m·a = r·m·ω 2 = r−1·m·v 2.
- The momentum of the body is p = m·v = r·m·ω.
- The moment of inertia is I = r 2·m.
- The angular momentum is L = r·m·v = r 2·m·ω = I·ω.
- The kinetic energy is E = 2−1·m·v 2 = 2−1·r 2·m·ω 2 = (2·m)−1·p 2 = 2−1·I·ω 2 = (2·I)−1·L 2 .
- The circumference of the orbit is 2·π·r.
- The period of the motion is T = 2·π·ω −1.
- The frequency is f = T −1 . (Frequency is also often denoted by the Greek letter ν, which however is almost indistinguishable from the letter v used here for velocity).
- The quantum number is J = 2·π·L h−1
[edit] Description of circular motion using polar coordinates
Figure 2: Polar coordinates for circular trajectory. On the left is a unit circle showing the changes

and

in the unit vectors

and

for a small increment
dθ in angle
θ.
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 handy 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:

[edit] Using complex numbers
Circular motion can be described using
complex numbers. Let the
x axis be the real axis and the
y axis be the imaginary axis. The position of the body can then be given as
z, a complex "vector":

where
i is the
imaginary unit, and

is the angle of the complex vector with the real axis and is a function of time
t. Since the radius is constant:

where a
dot indicates time differentiation. With this notation the velocity becomes:

and the acceleration becomes:


The first term is opposite to the direction of the displacement vector and the second is perpendicular to it, just like the earlier results shown before.