When an object is in uniform circular motion, the angular velocity and linear velocity of the object moving in a circle is constant.

**Centripetal Acceleration**, a

- centre-seeking
- is always perpendicular to the velocity
- always acts towards the centre of the circular motion

**Useful Equations: (Memorise)**

$a = \frac{v^{2}}{r}$

$a = r \omega^{2}$

$a = v \omega$

Since the centripetal acceleration is always directed towards the centre of the circular path, and the speed of the body is constant, the resultant force must also be directed towards the centre of the circle. The resultant force is centripetal force.

- Should not be drawn in as a force in free body diagrams.
- Does not perform any work when a particle moves in a circle because it is perpendicular to the displacement of the particle at any point in its motion.

**Useful Equations:**

$F = \frac{mv^{2}}{r}$

$F = mr \omega^{2}$

#### Mathematical Formulation

For uniform circular motion, the centripetal force ((F_c)) needed to keep an object moving in a circle with radius (r) at a speed (v) is given by (F_c = m\frac{v^2}{r}), where (m) is the mass of the object. This equation highlights the direct relationship between the force needed and the square of the speed of the object, as well as its inverse relationship with the radius of the circle.