Circular motion and centripetal force

Key ideas

• An object moving in a circle at constant speed is still accelerating, because its direction keeps changing.
• This centripetal acceleration always points toward the centre of the circle:

$a_c = \frac{v^2}{r} = \omega^2 r$

• The acceleration needs a net inward force, the centripetal force:

$F_c = \frac{mv^2}{r} = m\omega^2 r$

• Each variable, with sub-bullets:
  • $v$ — the speed around the circle (m s⁻¹)
  • $r$ — the radius of the circle (m)
  • $\omega$ — the angular velocity (rad s⁻¹), where $v = r\omega$
• Centripetal force is not a new force. It is the name for the net inward force, provided by real forces — tension, gravity, friction, the normal force, or a combination.
• For one trip around, $v = \dfrac{2\pi r}{T}$, where $T$ is the period.

Objects under two or more forces

At Level 3 the inward force is usually the resultant of two forces. Three classic set-ups:

• Conical pendulum — a mass swung in a horizontal circle on a string. The string tension and gravity combine; the horizontal part of the tension is the centripetal force:
  • vertical: $T\cos\theta = mg$
  • horizontal: $T\sin\theta = \dfrac{mv^2}{r}$
• Banked corner (no friction) — the road is tilted so the horizontal part of the normal force turns the car:

$\tan\theta = \frac{v^2}{rg}$

• Vertical circle — at the top, gravity and the normal (or tension) both point down toward the centre, so $N + mg = \dfrac{mv^2}{r}$. The slowest speed that keeps contact is when $N = 0$: $v_{min} = \sqrt{gr}$.

Tips

• Never add a separate "centripetal force" to a free-body diagram. Draw only the real forces (tension, weight, normal, friction), then set their resultant toward the centre equal to $\dfrac{mv^2}{r}$. Adding $F_c$ as an extra arrow double-counts it.
• Keep angles in the mode your calculator expects, and remember $v = r\omega$ lets you swap between speed and angular velocity whenever a question mixes them.