Radial Acceleration - Definition, Formula, Derivation, Units.

The object under motion can undergo a change in its speed. The measure of the rate of change in its speed along with direction with respect to time is called acceleration. The motion of the object can be linear or circular. Thus, the acceleration involved in linear motion is called linear acceleration. The acceleration involved in a circular motion is called angular acceleration.

Table of Contents

Radial Acceleration Definition
Radial Acceleration Formula
Tangential Component
Radial Acceleration Units
Frequently Asked Questions – FAQs

Introduction

If we recall circular motion, here distance covered will be along the curved path (circumference of a circle). The velocity involved is angular velocity and not linear velocity. Similarly, the acceleration involved is centripetal acceleration. Some important terms involved are-

Terms	Definition	Symbol	Formula	Units
Angular displacement	The angular difference between the initial and final position of an object.	𝜃	\(\begin{array}{l}\theta =\frac{s}{r}\end{array} \)	Radian
Angular velocity	The rate of change of angular displacement with respect to time.	⍵	\(\begin{array}{l}\omega =\frac{d\theta }{dt}\end{array} \)	radian/sec
Angular acceleration	The rate of change of angular velocity with respect to time.	𝛼	\(\begin{array}{l}\alpha =\frac{d\omega }{dt}\end{array} \)	radian/sec²

In a circular motion, the acceleration experienced by the body towards the centre is called the centripetal acceleration. This can be resolved into two-component. A radial component and a tangential component depend upon the type of motion.

Radial Acceleration Definition

In a uniform circular motion, “the acceleration of the object is along the radius, directed towards the centre” is called radial acceleration.

Radial Acceleration Formula

Schematic Representation of an Object under Rotational Motion

figure(1) schematic diagram representing an object under rotational motion

An object A is tied to a string and made to rotate about a fixed point O (centre) (see figure(1)). The object when rotated fast, the string (OA) looks almost like the radius of the circle, which implies that a force is exerted on the object from the centre, thereby an acceleration a₀ along the radial direction (along the radius of the circle towards the centre). To counter this force, tension is developed along the string in the opposite direction. This force due to tension is called centripetal force, thus the acceleration generated on the object is called centripetal acceleration or radial acceleration a_r.

Two positions of object A and B. They are infinitesimally close

figure(2) Positions of two objects A and B. They are infinitesimally close.

Centripetal Velocity Vector A and B

figure(3) The equivalent diagram of the centripetal velocity vector A and B.

Implementing the property of similar triangles we get-

\(\begin{array}{l}\frac{AB}{OA}=\frac{l}{r}\end{array} \)

Since A and B are infinitesimally close, we can approximate AB to the length of arc AB.

\(\begin{array}{l}AB=v\times dt\end{array} \)

From figure(3) since A and B are very close,

\(\begin{array}{l}v+dv\approx dv\end{array} \)

\(\begin{array}{l}\frac{AB}{OA}=\frac{dv}{v}\end{array} \)

\(\begin{array}{l}\frac{v\times dt}{r}=\frac{dv}{v}\end{array} \)

On rearranging,

\(\begin{array}{l}\frac{dv}{dt}=\frac{v^{2}}{r}\end{array} \)

Thus,

\(\begin{array}{l}\frac{dv}{dt}\end{array} \)

gives radial acceleration of an object under uniform circular motion. We arrive at the final expression-

\(\begin{array}{l}a_{r}=\frac{v^{2}}{r}\end{array} \)

Tangential Component

The tangential component is defined as the component of angular acceleration tangential to the circular path. The unit of measurement is m.s^-2. The mathematical representation is given as:

\(\begin{array}{l}a_{t}=\frac{v_{2}-v_{1}}{t}\end{array} \)

Where,

a_t is the tangential component
t is the time period
v₁ and v₂ are the respective velocities of the two objects in a circular motion

Radial Acceleration units

Radial Acceleration has the unit same as that of angular acceleration is radian/sec²

Physics Related Topics:

Angular Acceleration

Uniform accelerated motion

Differences Between Acceleration and Velocity

Understand uniform circular motion with the help of real-life examples.

Frequently Asked Questions – FAQs

What is acceleration?

Acceleration is defined as the rate of change of motion of a body. In other words, the measure of the rate of change in its speed along with direction with respect to time is called acceleration.

What are the main types of acceleration?

Linear acceleration and circular acceleration are the main types of acceleration.

What is linear acceleration?

The motion of the object can be linear or circular. Thus, the acceleration involved in linear motion is called linear acceleration.

What is circular acceleration?

The acceleration involved in a circular motion is called angular acceleration.

What is radial acceleration?

In a uniform circular motion, “the acceleration of the object is along the radius, directed towards the centre” is called radial acceleration.

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Radial Acceleration - Formula, Derivation, Units

Introduction

Angular displacement

Angular velocity

Angular acceleration