Uniform Circular Motion
Table of Contents
Uniform circular motion describes an object moving at constant speed along a circular path of radius \( r \). The direction of the velocity continuously changes, producing a radial (centripetal) acceleration toward the center, and the tangential acceleration is zero.
$$ a_c = \frac{v^2}{r} = \omega^2 r $$
Where
- \( \small a_c \) is the centripetal (radial) acceleration,
- \( \small v \) is the tangential speed,
- \( \small r \) is the radius of the circular path,
- \( \small \omega \) is the angular frequency,
- \( \small m \) is the mass,
- \( \small \theta \) is the angular position,
- \( \small T \) is the period (time for one revolution),
- \( \small f \) is the frequency, and
- \( \small s \) is the arc length along the path.
Related relationships #
Tangential (linear) speed:
$$ v = \omega r $$
Angular position as a function of time (constant \( \omega \)):
$$ \theta(t) = \theta_0 + \omega t $$
Period and frequency:
$$ T = \frac{2\pi}{\omega}, \quad f = \frac{1}{T}, \quad \omega = 2\pi f $$
Arc length and angle:
$$ s = r\theta $$
– Centripetal force (net radial force required for the circular path):
$$ F_c = ma_c = m\frac{v^2}{r} = m\omega^2 r $$