[Physics Class Notes] on Radial Acceleration Pdf for Exam

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. 

 

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:

Symbolically, these two units are written as 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:

 

 

This is for the centripetal or radial acceleration.

 

 

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.

 

 

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.

We  all know tha
t according to Newton’s Law Of Motion, any particular body or object has the tendency to change speed when in motion and this depends on the force and the amount of force that is put on the body or object. Acceleration is the measure of the rate of change in the speed and direction of the particular body or object, where the motion can be either linear or circular.

Linear acceleration refers to the type of acceleration that is involved in linear motion and  the type of acceleration that is  involved in circular motion is known as Angular acceleration.

The acceleration that is directed towards the center is known as Radial Acceleration and is measured in Radians per Square Second and the reason for acceleration is the Centripetal force.

When an object or a body moves with a non-uniform speed, it is tangential acceleration.

Features and Characteristics of Radial Acceleration:

1. Angular acceleration can be divided into Radial and Tangential acceleration

2. Radial acceleration shall be defined as an acceleration of an object that is directed towards the center.

3. If Radians per second square which is represented as ωs-2 is how Radial Acceleration is measured.

4. Radial acceleration is also known as Centripetal Acceleration.

5. The component of angular acceleration tangential to the circular path is what  Tangential Acceleration is.