• Mechanics
  • /
  • Kinematics
  • /
  • Circular motion

Circular motion is the movement whose trajectory describes a circular arc.

The figure shows the force that keeps the object in a circular path. It is the Earth's gravitational pull. This force originates at the satellite and always points to the Earth's center of gravity and acts as the centripetal force. This force is always perpendicular to the movement. It does not change the speed modulus. It only changes its direction.

Introduction to Circular Motion

Angular position \((\phi)\)
It is analogous to linear position \( s(t) \).
Unit: \( [\phi(t)]=rad \).
Angular velocity \((\omega)\)
It is the linear velocity analog \(v(t)\).
Unit: \( [\omega(t)] = \frac{rad}{s} \).
Angular acceleration \((\alpha)\)
It is analogous to linear acceleration \(a(t)\).
Unit: \( [\alpha(t)] = \frac{rad}{s^2} \).
Centripetal acceleration \((\vec{a}_{c})\)
It is the acceleration that points to the center of the circular trajectory. It keeps the body on the circular path.
Unit: \( [a_{c}]=\frac{m}{s^2} \) and \( a_{c} = r \omega^2 = \frac{v^2}{r}\).
Average angular speed \((\omega_m)\)
\(\omega_m = \frac{\Delta \phi}{\Delta t}\).
Unit: \([\omega_m]=\frac{rad}{s}\)
Average angular acceleration \((\alpha_m)\)
\(\alpha_m = \frac{\Delta \omega}{\Delta t}\).
Unit: \([\alpha_m]=\frac{m}{s^2}\)

All units are in the International System (\(IS\)).

We can relate this movement to a linear motion: \( s(t) = r\phi(t)\), \( v(t) = r\omega(t)\) and \( a(t) = r\alpha(t)\).

Uniform Circular Motion

It is the movement in which the angular velocity is constant and different from zero.

Time related functions in uniform circular motion: \begin{align} \omega(t) &= \omega_0 \notag \\ \phi(t) &= \phi_0 + \omega_0 t \notag \end{align}

To describe a uniform circular motion, these quantities are also used:

Period \((\tau)\)
Time interval spent by the object to make one complete revolution.
Frequency \((f)\) or Greaky letter \((\nu)\)
Number of repetitions per unit of time. Unit: Hertz \([f]=\frac{1}{seconds}=Hz\) .

Period and frequency relation: \( \tau = \frac{1}{f}\).
The angular velocity relation: \( \omega = \frac{2 \pi}{\tau} = 2 \pi f\).

Non-uniform Circular Motion

It is the movement in which the angular acceleration is constant and different from zero. Its equations are analogous to the ones on the uniformly accelerated motion: \begin{align} \phi(t) &= \phi_0 + \omega_0 t + \alpha \frac{t^2}{2} \notag \\ \omega(t) &= \omega_0 + \alpha t \notag \\ \omega^2 &= \omega_0^2 +2 \alpha \Delta \phi \notag \end{align}



Dynamic Exams
Differentiated Content
Top approval rate