The centripetal force is responsible for the centripetal motion of the object, and the centrifugal force acting against it prevents the object from meeting the centre of the circle.

**The centripetal acceleration and radius are obviously related to each other, as the object traveling in the circular track with the application of the centripetal force keeps the acceleration of the object along the radius of the circle. Both the vectors remain in a unique direction.**

**How is centripetal acceleration related to radius?**

The centripetal acceleration comes into the picture while the object is in a circular motion and elapses the path in a circle having specific radii.

**The object traveling in circular motion accelerates, maintaining a constant distance from the centre, which is the radius of the circular track. The direction of the centripetal acceleration of the object acts towards the centre of the circular path traced by an object along the radius of the circle.**

The force that keeps the object in a circular motion is a centripetal force. But this is not only responsible for the object moving along the circular path. The force acts opposing the centripetal force and prevents them from falling inward. The force that keeps them into place while traveling in a circular path is a centrifugal force.

**Find Centripetal Acceleration from Radius**

The object accelerating in the circular motion exerts a centripetal force. The centripetal force accelerating the object with velocity v is given by the expression:

F=mv^{2}/r

Here, m is the mass of the object, and

r is the radius of the circular path.

Since F=ma from Newton’s second law of motion. Using this in the above equation, we get:

ma=mv^{2}/r

Hence, the formula to find the centripetal acceleration of the object in a circular motion is given as:

a=mv^{2}/r

**According to this equation, the centripetal acceleration of the object is half of the square of the velocity of the object. The linear velocity is always perpendicular to the centripetal acceleration acting inward.**

The above equation clearly indicates that the angular acceleration is inversely related to the radius of the circular path. This implies that we shall have the highest centripetal acceleration of the object for the object propagating on small radii circles.

**Centripetal Acceleration and Radius Graph**

Now, let us understand the inverse relationship between the centripetal acceleration and the radius of the circular trajectory by working on one simple example.

Suppose different merry-go-rounds having varying radii are brought in a place to study the effect of the radius of the rotating wheel on the centripetal acceleration of the merry-go-round. The torque is applied on all the merry-go-rounds one by one maintaining the velocity of 3m/s constant.

For the first merry-go-round, the radius r=1 m, hence the centripetal acceleration for this wheel is,

a_{1}=mv^{2}/r

=

The second merry-go-round has a radius r=2 m, hence the centripetal acceleration for this wheel is,

a_{2}=v^{2}/r

The third merry-go-round has a radius r=3 m, hence the centripetal acceleration for this wheel is,

a_{3}=v^{2}/r

The fourth merry-go-round has a radius r=4 m, hence the centripetal acceleration for this wheel is,

a_{4}=v^{2}/r

The fifth merry-go-round has a radius r=5 m, hence the centripetal acceleration for this wheel is,

a_{5}=v^{2}/r

The data obtained is noted down in a below table:

No. of Merry-go-round | Radius (m) | Centripetal Acceleration (m/s^{2}) |

1^{st} | 1 | 3 |

2^{nd} | 2 | 1.5 |

3^{rd} | 3 | 1 |

4^{th} | 4 | 0.75 |

5^{th} | 5 | 0.6 |

Let us plot a graph of centripetal acceleration v/s radius for the above data.

From the above graph, we can say that the centripetal acceleration of the object in a circular motion decreases exponentially with the increasing radius. The magnitude of the centripetal acceleration decreases along the radius.

**Hence, to maintain the centripetal acceleration, the velocity of the object has to be increased as the circumference of the path traveled by the object increases.**

This is due to the fact that, as the radius increases, the centripetal force acting on the object reduces. We can relate this to the Coulomb force between the two unlike charges. As the linear spacing between the two increases, the magnitude of the force is reduced.

**What happens to centripetal acceleration if radius is doubled?**

The centripetal acceleration of the object is more for a small radius of the circle as compared to the larger radii.

**If the radius of the circular path is doubled, keeping the radial velocity of the object constant then, the centripetal acceleration of the object will be reduced to half.**

If the object is moving with a velocity ‘u’ along a circular path having a radius ‘r’, and the same object is moving on another circular track having a radius ‘2r’ with the velocity ‘v’, then the change in the centripetal acceleration is,

This equation gives the change in the centripetal acceleration of the object moving with different velocities while traveling on different circular tracks having a different radius.

**What is the centripetal acceleration of the car moving along the circular track on the stadium with a velocity of 20 km/h? The diameter of the stadium is 70 meters.**

**Given:** The velocity of the car is,

The diameter of the stadium is, d= 70 m.

Hence, the radius of the stadium is, r= 35 m.

The formula to calculate the centripetal acceleration of the car along the stadium is,

a=v^{2}/r

Substituting the values in this equation, we get,

Hence, the centripetal acceleration of a car is 0.86 m/s^{2} while driving around the stadium.

**What happens to the centripetal acceleration of the object after reducing the length of the rope to half to which it was attached at the length of 100 cm if the velocity of the object is doubled?**

**Given:** The initial length of the rope is, l_{1}=100 cm = 1 m

The final length of the rope is, l_{2}=100/2 cm = 50 cm = 0.5 m[/latex].

Let the initial velocity of the object be ‘u’ and the final velocity be ‘v’. The final velocity is doubled the initial velocity, hence, v=2u.

The initial centripetal acceleration of the object is,

The final centripetal acceleration of the object is,

Hence, we can see that the acceleration of the object increases 16 times more than the initial centripetal acceleration of the object after reducing the length of the rope to half.

**Conclusion**

The centripetal acceleration of the object moving in a circular track relies upon the radius of the circle. The centripetal acceleration is directed inward along the radius of the circle. Both are inversely related to each other. If the radius decreases, the centripetal acceleration of the object increases at an exponential rate.