If an object is moving along a circular path with a constant speed, it is indeed undergoing acceleration, despite the speed being constant. This is because acceleration is a vector quantity that includes both magnitude and direction.
In circular motion, even if the speed is constant, the direction of the velocity vector is constantly changing. Since velocity is a vector that incorporates both speed and direction, any change in direction indicates a change in velocity, and therefore acceleration.
The centripetal acceleration (aᵣ) is the acceleration experienced by an object moving in a circular path. It is always directed toward the center of the circle. The magnitude of centripetal acceleration can be calculated using the following formulas:
aᵣ = v²/r
where: aᵣ is the magnitude of centripetal acceleration, v is the speed of the object, and r is the radius of the circular path.
Furthermore, the angular velocity (ω) can be related to the linear speed (v) and the radius (r) through the equation:
v = ωr
Therefore, the centripetal acceleration can also be expressed as:
aᵣ = ω²r
So, even if the linear speed (v) is constant, there is still a centripetal acceleration (aᵣ) due to the change in direction, which is represented by the angular velocity (ω) and the radius (r) of the circular path.