# Angular And Tangential Acceleration: Relation, Diff, Conversion

In this article, the topic, “angular and tangential acceleration” with their Relationship, Difference, and Conversion will be discuss in a brief manner.Angular acceleration and tangential acceleration is not the same term.

Angular acceleration is not changes with the radius but tangential acceleration is changes with the radius. Angular acceleration can be explained as; changing in the angular velocity divided by the time and tangential acceleration can be explained as, changing in the linear velocity divided by the time.

### Angular acceleration:-

Angular acceleration is changes of the rate of time of angular velocity. The angular acceleration notified as radians cubic per second. The expression can be written as,

For the terms of the double derivation the angular acceleration can be written as,

Where,

α = Angular acceleration

dω = Changes rate of the angular velocity

dt = Changes rate of the time

The formula of angular acceleration is used to determine the angular acceleration and also its related facts.

### Tangential acceleration:-

The tangential acceleration can be explain as, the changing rate of tangential velocity of a substance in a particular circular way. The tangential acceleration can be notified as, meter per second square. The expression can be written as,

In term of distance the expression of tangential acceleration can be written as,

Or,

Where,

αt  = Tangential acceleration

v = Linear velocity

dv = Rate of the changing velocity

dt  = Rate of the changing time

ds = Rate of the changing covered distance

t = Time

The formula of tangential acceleration is used to determine the tangential acceleration and also its related facts.

## Is angular and tangential acceleration the same?

No, the meaning of the angular acceleration and tangential acceleration is not the same. Angular acceleration is not changes with the radius but tangential acceleration is changes with the radius. The tangential acceleration is the changing rate of tangential velocity of a substance in a particular circular way and angular acceleration is, change of time rate of angular velocity.

## Relation between angular acceleration and tangential acceleration:

When the value of angular velocity is remaining same the value of angular acceleration will be zero.

The Relation between angular acceleration and tangential acceleration is step by step describe below,

Let consider, a substance which is move through a circular path which radius will be r. The time will be taken by the substance will be Δt and the distance will be cover in an arc. The similar angle subtended is, Δθ.

When Δs writes the terms of Δθ the expression will be,

In a time Δt the expression can be write as,

In the limit of Δt the the expression will be,

The term of ds/dt represent as, linear speed which is tangential to the circle ω is angular speed.

Then the expression can be write as,

vr = rω …….eqn (4)

The above expression gives the connection in between linear speed and angular speed.

Eqn (4) only can be applicable for the motion which follows by the circular way. The connection in between linear speed and angular speed expression can be written as,

The two parameter $\vec{\omega }$ and $\vec{r}$ both are perpendicular to each other. If the eqn (4)  is differentiate respect to time then the expression can be written as,

dv/dt = r.dv/dt = ra

The expression dv/dt notified as tangential acceleration and expressed as, at = dω/dt is the angular acceleration α.

In that case the expression standing by, Image – Relation between angular acceleration and tangential acceleration

## Difference between angular acceleration and tangential acceleration:

The major difference in between the angular acceleration and tangential acceleration is listed below,

## How to find tangential acceleration from angular acceleration?

Let consider, a substance which is move through a circular path which radius will be r. The time will be taken by the substance will be Δt and the distance will be cover in an arc. The similar angle subtended is, Δθ.

When Δs writes the terms of Δθ the expression will be,

Δs = rΔθ …….eqn (1)

In a time Δt the expression can be write as,

In the limit of Δt the the expression will be,

The term of ds/dt represent as, linear speed which is tangential to the circle ω is angular speed.

Then the expression can be write as,

The above expression gives the connection in between linear speed and angular speed.

Eqn (4) only can be applicable for the motion which follows by the circular way. The connection in between linear speed and angular speed expression can be written as,

The two parameter $\vec{\omega }$ and $\vec{r}$ both are perpendicular to each other. If the eqn (4)  is differentiate respect to time then the expression can be written as,

The expression dv/dt notified as tangential acceleration and expressed as, at = dω/dt is the angular acceleration α.

In that case the expression standing by,

## How to find tangential acceleration without angular acceleration?

To find tangential acceleration without angular acceleration the process is describe below,

The expression can be written as,

In term of distance the expression of tangential acceleration can be written as,

Or,

Where,

αt  = Tangential acceleration

v = Linear velocity

dv = Rate of the changing velocity

dt  = Rate of the changing time

ds = Rate of the changing covered distance

t = Time

The formula of tangential acceleration is used to determine the tangential acceleration and also its related facts.

### Problem: –

A boat is going by a river from Dakshineswar temple to Belur math to follow a circular way. When the boat is going at that time the speed will be 30 meter per second to 30 meter per second in 70 second. Determine the acceleration to tangential.

Solution:-

Given parameters are listed below,

Initial speed of the boat = vi = 30 meter per second

Final speed of the boat = vf = 90 meter per second

Difference between the speed of the boat = dv = (90 – 30) meter per second = 60 meter per second

Initial time taken by the boat = ti= 30 sec

Final time taken by the boat = tf = 0 sec

Difference between the time taken by the boat = dt = (30 – 0) second = 30 second

From the formula of the tangential acceleration we can write,

at = dv/dt

at = 60/30

at = 30 meter per second square.

A boat is going by a river from Dakshineswar temple to Belur math to follow a circular way. When the boat is going at that time the speed will be 30 meter per second to 30 meter per second in 70 second. The acceleration to tangential for the boat is 30 meter per second square.

## Angle between tangential acceleration and angular acceleration:

The angle between the tangential and radial acceleration is always perpendicular to each other.

When an object moves in a circle, it has a centripetal acceleration directed toward the center of the circle.

We know that centripetal acceleration is given by

ac = v2/r

This centripetal acceleration is directed along a radius so it can be called the radial acceleration.
If the speed is not constant then there is also a tangential acceleration. The tangential acceleration is, indeed, tangent to the path of the particle performing motion.
Take the example of a turning rotor. Suppose the rotor is turning at a steady rate and there is no tangential acceleration, but there is centripetal acceleration. The point is following a circular path and its velocity (vector) is changing.

The direction it is pointing is changing every instant as it goes around the circle. Whenever the rotor is turning, every point on the rotor except the axis will have centripetal acceleration.
If the rotation rate of the rotor varies with time, then there is angular acceleration. If we look at a point on the rotor some distance r from the axis of the circle, then it will have a tangential acceleration along its circular path equal to r times the angular acceleration of the body.

Whenever the rotor as a whole has an angular acceleration. Every point on the rotor except points right on the axis of rotation will have a tangential acceleration
Hence, the angle between the tangential and radial acceleration is always perpendicular to each other.

## Angular acceleration to tangential acceleration:

The process of angular acceleration to tangential acceleration is discuss step after step,

Let consider, a substance which is move through a circular path which radius will be r. The time will be taken by the substance will be Δt and the distance will be cover in an arc. The similar angle subtended is, Δθ.

When Δs writes the terms of Δθ the expression will be,

Δs = rΔθ …….eqn (1)

In a time Δt the expression can be write as,

In the limit of Δt the the expression will be,

The term of ds/dt represent as, linear speed which is tangential to the circle ω is angular speed.

Then the expression can be write as,

The above expression gives the connection in between linear speed and angular speed.

Eqn (4) only can be applicable for the motion which follows by the circular way. The connection in between linear speed and angular speed expression can be written as,

The two parameter $\vec{\omega }$ and $\vec{r}$ both are perpendicular to each other. If the eqn (4)  is differentiate respect to time then the expression can be written as,

The expression dv/dt notified as tangential acceleration and expressed as, at = dω/dt is the angular acceleration α.

In that case the expression standing by,

at = rα …….eqn (6)

α = at/r…….eqn (7)

## Conclusion:

Angular acceleration and tangential acceleration is not the same term. Sometimes people are confused in between the angular acceleration and tangential acceleration but angular acceleration is rate of change in angular velocity and tangential acceleration is rate of change in linear velocity by time for both term.

Indrani Banerjee

Hi..I am Indrani Banerjee. I completed my bachelor's degree in mechanical engineering. I am an enthusiastic person and I am a person who is positive about every aspect of life. I like to read Books and listen to music. Let's connect through LinkedIn-https://www.linkedin.com/in/indrani-banerjee-2487b4214