Circular motion

Circular motion is where a particle moves in a circle. It occurs when the acceleration of the particle is constant in magnitude and always perpendicular to the velocity of the particle. As such, the speed of the particle remains constant as no work is done on the particle (the force acting on the particle is perpendicular to its displacement). The acceleration, $a$ of a particle undergoing circular motion may be describe as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a = - \omega^2 r}

where

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega} is the angular frequency, defined as $2\pi$ over the period of oscillation, $T$

$r$ is the radius of the orbit

Since $v=\omega r$, this may also be written as

$a={\frac {v^{2}}{r}}$

Example

Kepler's third law of planetary motion states that the square of the time period of a planet is proportional to the cube of the orbital distance. This can be seen as the earth undergoes circular motion around the sun. The force, $F$, acting on the earth is just gravity so

$F=-{\frac {GMm}{r^{2}}}$

Where

$G$ is Newton's gravitational constant

$M$ is the mass of the sun

$m$ is the mass of the earth

$r$ is the distance between the earth and the sun

From Newton's second law, this is can be related to the acceleration as:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{GMm}{r^2} = m \omega^2 r}

Since the angular speed, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega} , is equal to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2 \pi / T} where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T} is the time period, we can rewrite the previous equation as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T^2 = \frac{4 \pi^2}{GM} r^3}

This is Kepler's third law.

In reality, the orbits of the planets are not circular, instead they are elliptical. This derivation also assumes the mass of the earth is small compared to that of the sun. This is reasonable in this case but not for two stars that orbit as part of a binary system. When this more general case is accounted for, Kepler's third law becomes

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T^2 = \frac{4 \pi^2}{G(M+m)} a^3}

Where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} is the semi-major axis of the ellipse.