Radial Acceleration

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What is Radial Acceleration?

We know that when a body is subjected to an external force, it starts accelerating and this is what Newton’s second law says. Acceleration is nothing but the changing velocity of an object in a unit of time. It is a vector quantity that bears both magnitude and direction. It is measured in ms-2;however, one more term lies in Physics and that is Radial Acceleration.

So, do you know what radial acceleration is? Well! When angular velocity changes in a unit of time, it is a radial acceleration. 

Radial Acceleration

We know that a body can execute two types of motion and they are linear and circular motion. When it is linear motion, we consider just acceleration; however, during circular motion, which is actually an angular acceleration. 

We know that in a circular motion, the direction of the angular rate of velocity changes with time constantly so that’s why its angular acceleration gets two following components namely:

  • Radial acceleration

  • Tangential acceleration

So, let’s start with radial acceleration:

Radial acceleration is symbolized as ‘ar’ and it is the rate of change of angular velocity whose direction is towards the center about whose circumference, the body moves.

It happens because of the centripetal force. So centripetal force is the reason for a radial acceleration. 

A body whose mass is ‘m’ and the force acting on it is ‘mar’.....(1)

The formula for the centripetal force acting on the stone moving in a circular motion is mv2 /r….(2)

Equating (1) and (2):

       mar = mv2 /r

So, we get the radial acceleration formula as:

           ar = v2/r….(3)

Equation (3) is called centripetal acceleration

Radial Acceleration Units

The units of measurement are as follows:

  • Radians per second squared 

  • Meters per second squared

Symbolically, these two units are written as to ωs-2 or ms-2,respectively.

Radial Component

Let’s suppose that your child is on a merry-go-round. The direction of the velocity vector taken from your position will be tangential to the circular path in which the merry-go-round is making rounds. However, the centripetal acceleration points radially inwards or towards the center, which is what makes you go round. 

And from the formula in equation (3), we can see that the greater the radius of the circle of rotation, the lesser is its rate of change of velocity or the radial acceleration and vice-versa. Because of this reason we see that the smaller merry-go-rounds rotate a lot faster than the big ones. 

Now, it is crystal clear that the radial component is the primary reason for any object to keep making a circular motion.  

A body whose acceleration is always directed along the radius as its name signifies, there is one more component of acceleration whenever an object travels with a non-uniform speed and that is tangential acceleration ( at). The tangential acceleration acts tangentially to the path along which the object moves during a circular motion.

The below images show the variation of radial acceleration with the tangential acceleration:

(Image to be added soon)

This is for the centripetal or radial acceleration.

(Image to be added soon)

You can see the tangents drawn to the path of the object with the changing velocity.

We must keep in mind that the resultant acceleration is the sum of these two types of accelerations and the formula along with the required figure is stated below:

Formula: a = ar + at….(4)

Equation (4) states that the radial component of acceleration means the component of resultant acceleration which is perpendicular to the instantaneous velocity for the motion along any general path (not necessarily for circular motion). Since this component of acceleration is always directed along the radius of curvature of the trajectory (projectile motion), that's why the name radial acceleration is given to this type of acceleration.

(Image to be added soon)

Radial And Tangential Acceleration 

  • Radial acceleration is always along normal to the instantaneous velocity so it is also known as normal acceleration.

  • Radial acceleration is always directed towards the instantaneous center of curvature of the trajectory so it is also named centripetal acceleration.

  • Radial or centripetal acceleration is never defined only for circular motion, it may be defined for any type of motion.

  • The magnitude of radial acceleration at any instant is v2/r where v is the speed and r is the radius of curvature at an instant. In the case of circular motion, r will be the radius; also the direction of radial acceleration is along the radius of curvature.

  • The magnitude of the tangential acceleration is equal to the rate of change of speed of the particle w.r.t. time and it is always tangential to the path.

  • The tangential and normal accelerations are perpendicular components of the resultant acceleration so their vector sum returns the resultant acceleration.

  • In the case of uniform circular motion or UCM, tangential acceleration is always zero as speed doesn't change. In other words, the resultant acceleration vector in the case of UCM is orthogonal to the instantaneous velocity.

  • A body moving with a constant speed never bears any tangential acceleration regardless of the nature of the path.

  • For any rectilinear motion (be it uniform/non-uniform) radial acceleration is always zero. It is because the radius of curvature of a straight line is infinite.

  • A body moving along a curved trajectory will have some non-zero radial acceleration.

FAQ (Frequently Asked Questions)

1. What does an acceleration specify?

In mechanics, acceleration is the change of the velocity of an object with respect to time.

The orientation of the acceleration of the body is given by the alignment of the total force acting on that object. The magnitude of an object's acceleration as explained by Newton's Second Law is the combined effect of the following two causes:

  • The net balance of all external forces acting on the object’s magnitude varies directly with this net resulting force.

  • The object's mass depends on the materials out of which it is made and the magnitude varies inversely with the object's mass.

2. Are acceleration and deceleration the same?

In terms of Physics, both acceleration and deceleration are considered the same, because they both mean changes in velocity in a unit of time. Each of these accelerations viz: tangential, radial, and deceleration is felt by passengers until their velocity speed and direction matches with that of the uniformly moving car.

3. What does centripetal acceleration mean?

In a circular motion, the acceleration experienced by the body towards the center of the circle is called the centripetal acceleration. 

Acceleration can be resolved into two components viz: a radial component and a tangential component depending upon the type of motion an object makes.