Uniform circular motion

In physics, circular motion is a movement of an object along the circumference of a circle or rotation along a circular arc. It can be uniform, with a constant rate of rotation and constant tangential speed, or non-uniform with a changing rate of rotation. The rotation around a fixed axis of a three-dimensional body involves the circular motion of its parts. The equations of motion describe the movement of the center of mass of a body, which remains at a constant distance from the axis of rotation. In circular motion, the distance between the body and a fixed point on its surface remains the same, i.e., the body is assumed rigid.

Examples of circular motion include: special satellite orbits around the Earth (circular orbits), a ceiling fan's blades rotating around a hub, a stone that is tied to a rope and is being swung in circles, a car turning through a curve in a race track, an electron moving perpendicular to a uniform magnetic field, and a gear turning inside a mechanism.

Since the object's velocity vector is constantly changing direction, the moving object is undergoing acceleration by a centripetal force in the direction of the center of rotation. Without this acceleration, the object would move in a straight line, according to Newton's laws of motion...

Uniform circular motion

Figure 2: The velocity vectors at time $t$ and time $t + dt$ are moved from the orbit on the left to new positions where their tails coincide, on the right. Because the velocity is fixed in magnitude at $v = r ω$ , the velocity vectors also sweep out a circular path at angular rate $ω$ . As $dt \to 0$ , the acceleration vector $a$ becomes perpendicular to $v$ , which means it points toward the center of the orbit in the circle on the left. Angle $ω dt$ is the very small angle between the two velocities and tends to zero as $dt \to 0$ .

In physics, uniform circular motion describes the motion of a body traversing a circular path at a constant speed. Since the body describes circular motion, its distance 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 toward the axis of rotation. This acceleration is, in turn, produced by a centripetal force which is also constant in magnitude and directed toward 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.

Formulae

For motion in a circle of radius $r$ , the circumference of the circle is $C = 2 πr$ . If the period for one rotation is $T$ , the angular rate of rotation, also known as angular velocity, $ω$ is:

\omega ={\frac {2\pi }{T}}=2\pi f={\frac {d\theta }{dt}}

and the units are radians/second.

The speed of the object traveling the circle is:

v={\frac {2\pi r}{T}}=\omega r

The angle $θ$ swept out in a time $t$ is:

\theta =2\pi {\frac {t}{T}}=\omega t

The angular acceleration, $α$ , of the particle is:

\alpha ={\frac {d\omega }{dt}}

In the case of uniform circular motion, $α$ will be zero.

The acceleration due to change in the direction is:

a_{c}={\frac {v^{2}}{r}}=\omega ^{2}r

The centripetal and centrifugal force can also be found using acceleration:

F_{c}={\dot {p}}\mathrel {\overset {{\dot {m}}=0}{=}} ma_{c}={\frac {mv^{2}}{r}}

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

\mathbf {v} ={\boldsymbol {\omega }}\times \mathbf {r} ,

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

\mathbf {a} ={\boldsymbol {\omega }}\times \mathbf {v} ={\boldsymbol {\omega }}\times \left({\boldsymbol {\omega }}\times \mathbf {r} \right),

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]

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 1 metre per second.
The inward acceleration is 1 metre per square second, $v 2 / r$ .
It is subject to a centripetal force of 1 kilogram metre per square second, which is 1 newton.
The momentum of the body is 1 kg·m·s⁻¹.
The moment of inertia is 1 kg·m².
The angular momentum is 1 kg·m²·s⁻¹.
The kinetic energy is 1 joule.
The circumference of the orbit is 2 $π$ (~6.283) metres.
The period of the motion is 2 $π$ seconds per turn.
The frequency is (2 $π$ )⁻¹ hertz.

In polar coordinates

During circular motion, the body moves on a curve that can be described in the polar coordinate system as a fixed distance $R$ from the center of the orbit taken as the origin, oriented at an angle $θ (t)$ from some reference direction. See Figure 4. The displacement vector $\mathbf {r}$ is the radial vector from the origin to the particle location: