In the previous post, we briefly explained the relation between centripetal acceleration and gravity. This post is concerned with giving a detailed note on centripetal acceleration in uniform circular motion.

**A uniform circular motion means there will be constant speed while moving in a circular path. Then let us learn about centripetal acceleration with constant speed in a uniform circular motion in the following section.**

**What is centripetal acceleration in uniform circular motion?**

**The centripetal acceleration in uniform circular motion is a motion of an object whose acceleration is oriented towards the center of the rotational path at a constant speed and acts perpendicular to the instantaneous velocity.**

An interesting fact is that tangential acceleration is zero in a uniform circular motion as the angular velocity is constant. Still, centripetal acceleration is due to the orientation of the direction of the motion of the object.

**Is centripetal acceleration constant in uniform circular motion?**

**Uniform circular motion is concentrated only upon the centripetal components, which correspond to the object’s velocity and radius of the circular path. Since the radius is constant in a uniform circular motion and even velocity change is constant, centripetal acceleration is not constant.**

Since we know that in a uniform circular motion, the speed is always constant, only the magnitude of the motion is constant and not the direction. Thus, it is said that we can achieve constant linear acceleration in uniform circular motion but not centripetal acceleration.

In another aspect, the centripetal acceleration is caused by the centripetal force exerted on the object under circular motion. If the centripetal force is kept constant, then centripetal acceleration in uniform circular motion will be constant, which indeed hard to achieve. The centripetal force is constant only if there is vertical circular motion.

**Why centripetal acceleration is not constant in uniform circular motion?**

**Even though the velocity and radius are constant, the centripetal acceleration is not constant because the centripetal acceleration is mainly associated with the direction.**

The constant velocity and radius refer to only the magnitude of the motion are constant but not the direction. Every circular motion is associated with two components tangential and centripetal components. The tangential components always depend on the magnitude of the motion, and centripetal components are associated with the direction.

Thus, we can observe constant tangential acceleration as velocity and radius exhibit tangential behavior but the centripetal acceleration in uniform circular motion is not constant.

**Find centripetal acceleration in uniform circular motion**

**The centripetal acceleration can also be stated as the rate of change of tangential velocity of a body under circular motion. Thus, to find centripetal acceleration, the tangential velocity is required. Since the body is under circular motion, the radius of the circular orbit is also essential for the process.**

Let us consider the motion of a body of mass ‘m’ under uniform circular motion whose velocity v_{1} changes to v_{2} to produce acceleration. The radius of the circular path is ‘r’. To retain the motion of body in a circular path, a small amount of force is required, so using Newton’s second law of motion, force can be written as F=ma. Since force is directed toward the center of the rotation axis, the force is called centripetal force, and the resultant acceleration is centripetal.

The change in velocity can be written as Δv=v_{2}-v_{1}.

But we know that acceleration of a body under any motion is given by

[latex]a=\frac{\Delta v}{\Delta t}[/latex]

In order to solve, let us consider a diagram given below.

From the above diagram, consider ΔOAB and ΔXYZ which are similar. From both the triangle

[latex]\frac{\Delta v}{AB}=\frac{v}{r}[/latex]

But AB=vΔt

[latex]\frac{\Delta v}{v\Delta t}=\frac{v}{r}[/latex]

Rearranging the terms, we get

[latex]\frac{\Delta v}{\Delta t}=\frac{v^2}{r}[/latex]

We can rewrite the above equation from the first equation as centripetal acceleration.

[latex]a_c=\frac{v^2}{r}[/latex]

But the velocity v can be written as v=ωr, where ω is the angular velocity. Substituting the value we get,

[latex]a_c=\frac{(\omega r)^2}{r}[/latex]

a_{c}=ω^{2}r

**The direction of centripetal acceleration in uniform circular motion**

**The direction of the centripetal acceleration in uniform circular motion is always in the inward direction pointed towards the center of the circle. Just is similar to the planetary motion where the sun is at the center and the motion of the planet is concentrated towards the sun.**

For example, we know that gravity works as a centripetal force. When an object is set to fall freely from the space, the centripetal acceleration is oriented towards the center of gravity, and thus the object is pulled down towards the earth.

Many people think that centripetal acceleration acts tangentially outward, which is wrong. The centripetal acceleration in uniform circular motion is oriented radially inward. **Centripetal acceleration is the central acceleration that is essential to put the object to move along the circular axis. If the acceleration points outward, then the object may escape from the circular path.**

**Solved problems on centripetal acceleration in a uniform circular motion**

**A disc of mass 8kg is rotating at 12m/s in a circular axis of radius 3m. Calculate the disc’s centripetal acceleration and find the centripetal force acting on the disc to remain in the circular axis.**

**Solution:**

Given –mass of the disk m=8kg

Velocity of the disc v=12m/s

Radius of the circular path r=3m.

The centripetal acceleration is given by

[latex]a_c=\frac{v^2}{r}[/latex]

[latex]a_c=\frac{12^2}{3}[/latex]

a_{c}=48m/s^{2}

The centripetal force F_{c}=ma_{c}

F_{c}=(12×48)

F_{c}=576 N.

**Calculate the centripetal acceleration of an object of mass 4kg rotating with an angular speed is 3m/s and radius of rotational axis is 8m. And also find the linear velocity and force exerted to keep the body in rotation.**

**Solution:**

Given –the angular speed of the object ω=3m/s

The radius of the rotational axis r=8m.

Mass of the object m=4kg.

The centripetal acceleration a_{c}=ω^{2}r

a_{c}=(3)2(8)

a_{c}=72m/s^{2}

The centripetal force is F_{c}=ma_{c}

F_{c}=(4×72)

F_{c}=288 N.

The linear velocity v=ωr

v=3×8

v=24m/s.

**The centripetal force exerted on a body of mass 45kg is 583N. Calculate the centripetal acceleration and velocity of the body. The radius of the rotational axis is 16m.**

**Solution:**

Given –the mass of the body, m=45kg

The radius of the rotational axis r=16m.

Exerted centripetal force F_{c}=583N.

The centripetal acceleration can be calculated using Newton’s second law F_{c}=ma_{c}

Rearranging the equation, [latex]a_c=\frac{F_c}{m}[/latex]

[latex]a_c=\frac{583}{45}[/latex]

a_{c}=12.95m/s^{2}.

Velocity of the body is given by rearranging the centripetal acceleration equation

[latex]a_c=\frac{v^2}{r}[/latex]

v^{2}=a_{c}r

[latex]v=\sqrt{a_cr}[/latex]

[latex]v=\sqrt{12.95\times16}[/latex]

[latex]v=\sqrt{207.2}[/latex]

v=14.99m/s.

**Conclusion**

In this post, we learned that centripetal acceleration in uniform circular motion is a variable entity often defined as the change in the direction associated with the object under motion in a circular orbit.