Motion in a Circle

• $\theta=$$\frac{s}{r}$ (by name)


  $\theta$= Angular Displacement

  s = Arc Length

  r = Radius
• One radian is the angle subtended at the centre of the circle by an arc whose arc length is equal to the radius of the circle.
• Angular velocity is the rate of change of angular displacement. Its unit is rad s^-1.
• $\omega=$$\frac{\theta}{t}$ (by name)


  $\omega$= Angular velocity

  $\theta$= Angular Displacement

  t = Time taken
• If a body undergoing uniform circular motion takes T seconds to complete 1 full revolution, then

  $\omega=$$2\pi{f}$

  where f is the frequency of the circular motion.
• $v=$$r\omega$ (in units)


  v = m s^-1

  r = m

  $\omega$= rad s^-1
• The tangential speed of any point is proportional to its distance from the axis of rotation.
• $a=$$\frac{v^2}{r}=$$r\omega^2=$$v\omega$ (in units)


  a = m s^-2

  v = m s^-1

  r = m

  $\omega$= rad s^-1
• Centripetal acceleration is the acceleration of a body undergoing uniform circular motion. Its direction is towards the centre of the circular path.
• Centripetal acceleration is always perpendicular to the velocity.
• By Newton's 2nd Law of Motion, a body undergoing uniform circular velocity would experience a force acting towards the centre of the circular path due to the centripetal acceleration directing towards the centre. This force is known as the centripetal force.
• $F_c=$$m\frac{v^2}{r}=$$mr\omega^2=$$mv\omega$ (by name)


  $F_c$= Centripetal Force

  m = Mass

  v = Linear Velocity

  r = Radius

  $\omega$= Angular Velocity
• The fact that centripetal force is normal to the velocity means that work is not being done by this force as displacement of the object is always perpendicular to the force.
• $tan \ \theta=$$\frac{v^2}{gr}$ (in units)


  v = m s^-1

  g = m s^-2

  r = m