Centripetal acceleration is the acceleration that is directed towards the center of a circular path for an object moving in a circular motion. It is always perpendicular to the velocity of the object and is responsible for changing the direction of the object's velocity while it moves along the circular path.

To derive the expression for centripetal acceleration, we can start with the formula for acceleration:
\[ a = \frac{v^2}{r} \]
Where:
- \( a \) = acceleration
- \( v \) = velocity of the object
- \( r \) = radius of the circular path
Since the velocity of an object moving in a circular path is tangential to the circle at any given point, the acceleration must point towards the center of the circle. This acceleration is the centripetal acceleration.
Therefore, the centripetal acceleration can be expressed as:
\[ a_c = \frac{v^2}{r} \]

- Centripetal acceleration is essential for an object to maintain its circular motion.
- The magnitude of centripetal acceleration is directly proportional to the square of the velocity and inversely proportional to the radius of the circular path.
- It is responsible for the change in direction of the object's velocity, making it curve along the circular path.
- Without centripetal acceleration, the object would move in a straight line tangent to the circle instead of following the circular path.
- Centripetal acceleration plays a crucial role in various phenomena such as planetary orbits, carousel rides, and even the motion of vehicles around curves.