Circular motion

Uniform circular motion of radius

a

and angular frequency

ω

has centripetal acceleration of magnitude

a ω^{2}

toward the centre.

Table of contents
- Centripetal acceleration Circular motion
- Circular orbit Circular motion

Centripetal acceleration ( $a = - ω^{2} r$ )

For uniform circular motion of radius

R

and angular speed

ω

, the acceleration is directed toward the centre and has magnitude

ω^{2} R = v^{2} / R

r

for the position relative to the centre gives

\ddot{r} = - ω^{2} r

. The required resultant force is

m \ddot{r}

; it may be supplied by a normal force, gravity, or another physical interaction.

Circular orbit

A circular orbit maintains constant distance from its centre of attraction. Its inward radial acceleration has magnitude

v^{2} / r

, so the radial force must satisfy

F_{r} = - m v^{2} / r

.

Circular orbits in an exponential central potential ( $l^{2} = r^{3} e^{- r}$ )

For unit mass with specific angular momentum

l

in

V (r) = - e^{- r}

, the effective potential is

- e^{- r} + l^{2} / (2 r^{2})

. A positive-radius circular orbit satisfies the displayed equation. Since

r^{3} e^{- r}

has its unique maximum

27/ e^{3}

at

r = 3

, exactly two radii occur for

0 < l^{2} < 27/ e^{3}

. Their radial stability follows from

V_{eff}^{''} = e^{- r} (3/ r - 1)

: the inner orbit is stable at fixed angular momentum, the outer unstable. The case

l = 0

has no finite-radius circular orbit despite satisfying the upper bound.

Circular-orbit stability for a power-law central force ( $Ω_{r}^{2} = (n + 3) k a^{n - 1}$ )

For the radial acceleration

- k r^{n}

with

k > 0

, a circular orbit of radius

a

has squared specific angular momentum

h^{2} = k a^{n + 3}

. At fixed

h

, its radial small oscillations have squared frequency

(n + 3) k a^{n - 1}

, so the orbit is stable for

n > - 3

. This is a special case of the effective potential stability criterion.

Circular orbit in a quadratic central potential ( $r_{0}^{4} = h^{2} / (2 k m^{2})$ )

For

V (r) = km r^{2}

, fixed nonzero angular momentum

h

gives the effective potential

V_{eff} = h^{2} / (2 m r^{2}) + km r^{2}

. A positive-radius circular orbit exists when

k > 0

, at the displayed radius, and

V_{eff}^{''} (r_{0}) = 8 km > 0

proves radial stability by the effective potential stability criterion. Its angular speed has magnitude

2 k

and its small radial oscillation frequency is

8 k

. If

k \leq 0

, no such orbit with nonzero angular momentum exists; for

h = 0, k > 0

, the sole minimum is at the origin, not a positive-radius circular orbit.

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Circular motion by

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Circular motion refers to the movement of an object along the circumference of a circular path. This type of motion can be classified into two main categories: 1. **Uniform Circular Motion (UCM)**: In uniform circular motion, an object moves around a circle at a constant speed. Although the speed remains constant, the velocity is constantly changing because the direction of the object's motion is continually changing. This is due to centripetal acceleration, which acts towards the center of the circular path.

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