Circular Motion

Angular velocity, centripetal acceleration, and centripetal force in uniform circular motion.

Spec Points Covered
  • I can define angular velocityThe rate of change of displacement. A vector quantity. Measured in m s⁻¹.The rate of change of angular displacementThe distance moved in a particular direction from a starting point. A vector quantity. Measured in metres (m).. The angle swept per unit time for an object moving in a circle. Measured in rad s⁻¹. and convert between angular velocityThe rate of change of displacement. A vector quantity. Measured in m s⁻¹.The rate of change of angular displacementThe distance moved in a particular direction from a starting point. A vector quantity. Measured in metres (m).. The angle swept per unit time for an object moving in a circle. Measured in rad s⁻¹., frequencyThe number of complete oscillations passing a point per unit time. Measured in hertz (Hz)., and periodThe time taken for one complete oscillation or wave cycle. Measured in seconds (s)..
  • I can $use \omega =$ $2\pi/$T $and \omega =$ $2\pi$f to solve problems.
  • I can relate linear speed to angular velocityThe rate of change of displacement. A vector quantity. Measured in m s⁻¹.The rate of change of angular displacementThe distance moved in a particular direction from a starting point. A vector quantity. Measured in metres (m).. The angle swept per unit time for an object moving in a circle. Measured in rad s⁻¹. using v = $r\omega.$
  • I can explain why an object moving in a circle at constant speed is accelerating.
  • I can calculate centripetal accelerationThe rate of change of velocity. A vector quantity. Measured in m s⁻².The accelerationThe rate of change of velocity. A vector quantity. Measured in m s⁻². directed towards the centre of the circular path, required to maintain circular motion. using a = \(v^{2}\)/r and a = $r\omega$^2.
  • I can calculate centripetal forceThe resultant force directed towards the centre of a circular path that causes an object to move in a circle. It is not a separate force but the net force providing circular motion. using F = mv^2/r and F = $mr\omega$^2.
  • I can identify the physical force providing the centripetal forceThe resultant force directed towards the centre of a circular path that causes an object to move in a circle. It is not a separate force but the net force providing circular motion. in different situations (friction, tension, gravity, normal contact).
  • I can analyse circular motion problems involving cars on flat roads and banked tracks.
  • I can analyse the conical pendulum and determine the tension and angle.
  • I can analyse vertical circular motion and determine conditions for maintaining contact at the top of a loop.
Notes
01 Angular velocity (\omega) Angular velocity (\omega) → 02 This equation links the rotational description ($\omega$) to the translational description (v) $v = r\omega$ → 03 Centripetal acceleration Centripetal acceleration → 04 Centripetal force is not a new type of force $F = \frac{mv^2}{r} = mr\omega^2$ → 05 On a flat road → 06 A conical pendulum is a mass on a string that traces a horizontal circle, with the string → 07 In a vertical circle → 08 Satellite orbiting Earth →
Σ Key Equations Full Reference →
On Data Sheet
Not on Data Sheet
Angular velocity from period
$$\omega = \frac{2\pi}{T}$$
  • Where:
    • $\omega$ = rad \(s^{-1}\)
    • $T$ = s
Use when given the period of rotation. Also on the data sheet.
Linear speed from angular velocity
$$v = r\omega$$
  • Where:
    • $v$ = m \(s^{-1}\)
    • $r$ = m
    • $\omega$ = rad \(s^{-1}\)
Links translational and rotational descriptions. Points further from the axis have greater v for the same \omega.
Centripetal acceleration (speed form)
$$a = \frac{v^2}{r}$$
  • Where:
    • $a$ = m \(s^{-2}\)
    • $v$ = m \(s^{-1}\)
    • $r$ = m
Use when given linear speed. Always directed towards the centre.
Centripetal acceleration (angular form)
$$a = r\omega^2$$
  • Where:
    • $a$ = m \(s^{-2}\)
    • $r$ = m
    • $\omega$ = rad \(s^{-1}\)
Use when given angular velocity. Equivalent to \(v^{2}\)/r via v = r\omega.
Centripetal force (speed form)
$$F = \frac{mv^2}{r}$$
  • Where:
    • $F$ = N
    • $m$ = kg
    • $v$ = m \(s^{-1}\)
    • $r$ = m
The resultant force towards the centre required for circular motion. Not a new type of force.
Centripetal force (angular form)
$$F = mr\omega^2$$
  • Where:
    • $F$ = N
    • $m$ = kg
    • $r$ = m
    • $\omega$ = rad \(s^{-1}\)
Alternative form using angular velocity.
Angular velocity from frequency
$$\omega = 2\pi f$$
  • Where:
    • $\omega$ = rad \(s^{-1}\)
    • $f$ = Hz
Must memorise. Not explicitly on the data sheet in this form but follows directly from \omega = 2\pi/T and f = 1/T.