Radial Acceleration: Definition, Derivation, Formula and Units

Radial Acceleration includes its main usage in terms of movement, velocity and direction of the objects. All these phenomena are measured in the relative context. The acceleration could also be defined as a changing velocity of any object at any particular time. Acceleration is referred to as a vector quantity that confines in itself both magnitude and the direction.

What is Radial Acceleration?

As per Newton's law of motion, any object or body which is under motion tends to undergo a change in its speed through movement and this varies on the basis of the amount of force applied to an object. 

Therefore, the measure of the rate of change in the speed of an object along with that of its direction and with respect to time is called acceleration. Although, the motion of the object can be either linear or circular.

The type of acceleration that is involved in linear motion is called linear acceleration. While the type of acceleration that is involved in a circular motion is called angular acceleration. The angular acceleration can be classified as follows:

• Radial Acceleration
• Tangential Acceleration

Radial acceleration shall be defined as an acceleration of an object that is directed towards the centre. This acceleration takes place in a uniform circular motion and the movement is concerned along the radius of an object. 

On the other hand, tangential acceleration measures the rate of change of tangential velocity of a point at a certain radius with respect to time. Tangential acceleration is similar to linear acceleration but is relevant to circular motion.

Units of the Radial Acceleration

Radial acceleration is measured in terms of Radians per second square which is represented as ωs^-2

Formulas Related to Radial Acceleration

In the above figure, an object M is tied to a string and then it is made to rotate at a fixed axis around the point ‘C’ which is the centre of the circle. The object when rotated fast, the string CM appears like the radius of the circle. A force is exerted on the object from the centre, thereby an acceleration a₀ acts along the radial direction, that is, along the radius of the circle towards the centre.

In order to counter this force, the tension is developed along the string in the opposite direction. This force which emerges due to the tension is called centripetal force. And the acceleration that is generated on an object is called the centripetal acceleration or radial acceleration a_r.

The above figure represents the two objects which are infinitesimally closer to each other and the specific diagram of an equivalent triangle having a centripetal velocity vector.

On applying the property of the similar triangles, it could be seen that 

\(\frac {AB} {OA} = IR\) 

Since, the points A and B are infinitesimally closer, it could be assumed as AB to the length of arc AB as 

AB = v × dt

The points A and B are very close, therefore,

v + dv ≈ dv × \(\frac {AB} {OA}\)

→ \(\frac {dv} {v}\) ×\(\frac {v x dt} {r}\)

 → \(\frac {dv} {v}\)

Now, on rearranging,

\(\frac {dv} {v} = \frac {v^2} {r}\)

Hence, dv / dt represents the radial acceleration of an object which is under the uniform circular motion. Therefore, the final equation for the above proving is given below.

a_r = \(\frac {v^2} {r}\)

Tangential Acceleration

The tangential acceleration is generally defined as the component of angular acceleration tangential to the circular path. The unit of measurement of the tangential acceleration is represented as ms^-2. And the mathematical equation for the same is given as:

a_t = \(\frac {v_2 - v_1} {t}\)

Here,

a_t → The tangential component

t → Time period

v₁ and v₂ → Respective velocities of the two objects which are given in a circular motion

Things to Remember based on Radial Acceleration

• Angular acceleration can be classified into Radial and Tangential acceleration
• Radial acceleration shall be defined as an acceleration of an object that is directed towards the centre.
• Radial acceleration is measured in terms of Radians per second square which is represented as ωs^-2 
• Centripetal acceleration is also called radial acceleration.
• The tangential acceleration is generally defined as the component of angular acceleration tangential to the circular path.