When an object moves in a circular trajectory, it experiences a continuous change in its direction of motion. This change in direction, despite constant speed, indicates the presence of acceleration. This acceleration, directed toward the center of the circular path, is called centripetal acceleration.
Circular Motion:-
When an object moves along a circular path with constant speed, its velocity is constantly changing in direction due to the continuous curvature of the path. Velocity is a vector quantity, meaning it has both magnitude and direction. Even if the speed remains unchanged, the change in direction contributes to acceleration.
Two main types of circular motion:
1.Uniform Circular Motion (UCM): Object moves with a constant speed along a circular path.
2.Non-Uniform Circular Motion: Object’s speed varies as it moves along the circular path.
Centripetal acceleration can be mainly discussed in the situation of uniform circular motion, where the speed of the object remains constant, but the direction of velocity changes continuously.
What is Centripetal Acceleration?
Centripetal acceleration refers to the acceleration experienced by an object moving in a circular path, directed towards the center of the circle.
Term “centripetal” means “center-seeking,” and the force or acceleration always points towards the center of the circular path. Inward acceleration is what keeps the object moving along the curved path quite than in a straight line (as per Newton’s First Law of Motion, an object in motion tends to stay in motion in a straight line unless acted upon by an external force).
Derivation and Formula of Centripetal Acceleration:
To derive the formula for centripetal acceleration, consider an object of mass m moving with a constant speed v along a circular path of radius r.
Velocity of the object constantly changes its direction as the object moves, while the magnitude of velocity (speed) remains the same. Change in velocity over a short time interval Δt gives rise to acceleration. Since this change in velocity is due to the change in direction, and not magnitude, Acceleration must be perpendicular to the velocity, pointing towards the center of the circle.
Formula for centripetal acceleration ac, can be derived as: ac = v2 / r
Here,
ac is the centripetal acceleration,
v is the speed of the object, and
r is the radius of the circular path.
Formula indicates that centripetal acceleration is directly proportional to the square of the speed and inversely proportional to the radius of the path. For a given speed, a smaller circular path results in a higher centripetal acceleration, and for a given radius, a higher speed leads to greater centripetal acceleration.
Characteristics of Centripetal Acceleration:-
1.Direction: Direction of centripetal acceleration is always towards the center of the circular path.
2.Magnitude: Magnitude of centripetal acceleration depends on both the speed of the object and the radius of the path, as given by the formula
ac = v2 / r
3.Cause: Centripetal acceleration is caused by a centripetal force, which is the net force acting towards the center. This force can be tension, gravitational force, friction, or any other force that causes the circular motion.
4.Non-tangential Nature: Centripetal acceleration is not along the tangent of the circular path but perpendicular to it, directed inward.
Centripetal Force and Relation with Centripetal Acceleration:-
Centripetal acceleration is produced by a centripetal force. According to Newton’s Second Law of Motion, the force acting on an object is the product of its mass and acceleration. Therefore, the centripetal force Fc is given by:
Fc = m⋅ ac = m⋅v2 / r
Where,
m is the mass of the object,
ac is the centripetal acceleration,
v is the speed of the object, and
r is the radius of the circular path.
Relation helps to understand that for an object moving in a circle, centripetal force is what keeps it in motion along that path, and it is directly responsible for the centripetal acceleration.
Examples of Centripetal Acceleration:-
1.Earth’s Orbit around the Sun:-The Earth, moving around the Sun, experiences centripetal acceleration due to the gravitational force exerted by the Sun.
2. Car Taking a Turn: A car moving along a curved road experiences centripetal acceleration, which is provided by the friction between the tires and the road.
3. Swinging a Ball on a String: When we swing a ball tied to a string in a circular motion, the tension in the string provides the centripetal force, causing centripetal acceleration
Note:-
Centripetal acceleration provides insights into how objects maintain their curved paths and what forces are responsible for keeping them in motion.