Centripetal Acceleration

With regards to kinematics, acceleration refers to the change in speed or velocity, both in its direction and magnitude, or in either of them. Moreover, the direction of velocity alters continuously in case of uniform circular motion. Hence, a related acceleration is consistently present even if the speed may be fixed. This acceleration of a body moving in uniform circular motion is called centripetal acceleration.

A centripetal acceleration example can be that of taking a sharp turn in a vehicle. If a person is holding the wheel constant while taking a turn, he/she is in a uniform circular motion.

Derivation Of Centripetal Acceleration Formula

The following diagram represents a body moving in a circular pathway at a persistent speed.

Here, centripetal acceleration direction is pointed towards the centre of curvature. Furthermore, you must note that triangles created by radii r, Δs and velocity vectors are the same. Additionally, triangles PQR and ABC have two identical sides which mean they are isosceles triangles. Moreover, these two same sides of speed vector triangles are velocities v1 = v2 = v. Then, by utilising the features of two equal triangles, it is found:

• Δv / v = Δs / r

Acceleration is Δv / Δt, and solution for Δv is:

• Δv = ( v / r )Δs

Further by dividing by Δt,

• Δv / Δt = v / r × Δs / Δt

So finally, note that

 Δv / Δt = ac, and Δs / Δt = v, tangential or linear speed, the magnitude of centripetal acceleration is ac = v2 / r

So, with this equation, you can determine that centripetal acceleration is more significant at high speeds and in smaller radius curves.

Note: The S.I unit for centripetal acceleration is m/s2

Solved Numericals

1. A rock tied to a string is moving at a fixed speed of 10.0 m/s in a circle having radius 8.0 m. Evaluate the approximate magnitude of centripetal acceleration of the rock.

Solution. According to the formula ac = v2 / r

Therefore, ac = (10)2/ 8

                    ac = 12.5 m/s2

2. In the case of a slot-car set, its maximum centripetal acceleration without being ejected from its track is noted to be 3.8 meters per second squared. It is noticed that these slot cars fly off their track when they exceed 1.1 meters per second. What is the radius of the curve in the track? Answer in metres.

Solution. Maximum possible centripetal acceleration is a = 3.8 m/s2, and maximum speed which can be attained by these slot cars without flying off its track is 1.1 m/s.

Applying this centripetal acceleration formula the answer is:

ac = v2 / r

Therefore, r = v2 / ac

                     = (1.1m/s)2 / 3.8 m/s2

                     = 0.32m

Do It Yourself

1. An Object is Travelling in a Circle at a Constant Speed.

(a) It has a constant velocity 

(b) The body has no acceleration 

(c) It has an inward acceleration 

(d) The body has an outward radial acceleration

2. For a Particle Performing a Uniform Circular Motion, the Acceleration is:

(a) constant in direction

(b) constant in its magnitude but not in the direction

(c) constant in its magnitude and direction

(d) constant in neither its magnitude nor direction

3. When a Particle is Revolving with Uniform Speed on a Given Circular Path:

(a) zero force is acting on it

(b) no acceleration is acting on it

(c) zero work is being done by it

(d) it has a velocity which is constant

By going through this material, you must have understood what is meant by centripetal acceleration. Additionally, for more concepts and ideas related to Physics, you can also download our Vedantu app and get access to a vast range of study materials along with online tutorials.