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Tangential Acceleration

At any point on a circle, you can pick two special directions: The direction that points directly away from the center of the circle (along the radius) is called radial direction, and the direction that is perpendicular to this is called the tangential direction.

Consider a particle which is moving along a curved or circular. When an object moves in a curved path, the magnitude and direction of the velocity of the object change. Direction of the velocity at any point on the circular path is in the direction of the tangent at that point. But direction of the acceleration at any point is not in the same direction of the tangent at that point,
the acceleration vector $\vec{a}$, however, is at sum angle to the path.

At each instant, the particle can be modeled as if it were moving on a circular path. The radius of the circular path is the radius of curvature of path at that instant. At the next instant, the particle is moving as if on a different circular path, with different center and a different radius than the previous one. At each of three points A,B,C in figure below, we see the dashed circles that from geometric models of circular paths for the actual path at each point.

Tangential Acceleration
Figure 1: Tangential acceleration (at) can be explained as the linear acceleration of a particle at any point on the curved path.

The acceleration can consist of two components one is the radial acceleration vector represented by ar and other is tangential component of acceleration at. Acceleration at any point in the curved point is equal to vector sum of tangential and radial acceleration. Path of the particle is shown above in the Figure 1. Direction of the acceleration vector is also shown which is at any point on the curved path is the resultant vector of vector ar and at. Magnitude of 'a' is given below,
a = $(a_{r}^2 + a_{t}^2)^\frac{1}{2}$

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What is Tangential Acceleration?

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Tangential acceleration is a measure of how the tangential velocity of a point at a certain radius changes with time. Tangential acceleration is just like linear acceleration, but it’s particular to the tangential direction, which is relevant to circular motion.

Tangential Acceleration Formula

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To derive the formula for the tangential acceleration we first derive the concept of the Tangential speed and then Tangential acceleration formula is derived using it.

It results in the change of the speed of the object along the curved path and at a particular point on the curved path it is equal to the instantaneous velocity at that particular point.

$a_{t}$ = $\frac{d\left |v \right |}{dt}$

Radial acceleration is because of the change in the direction of the velocity. Its magnitude is equal to tangential acceleration equation is given by,

| $a_{r}$ | = $\frac{v^{2}}{r}$

r = radius of curvature

Tangential Acceleration units:
SI unit of tangential acceleration is m/sec2.

Tangential Speed

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Tangential Speed is equal to the instantaneous linear speed of the particle at any point on the curved path. Tangential speed at any point is equal to the multiplication of the radius of curvature and angular speed at a particular point on the curved path. Angular speed is defined as the time rate of change of the angular displacement along the curved path.
$V_{t}$ = r$\omega$
Where,
r = radius of curvature
$\omega$ = angular speed at a particular point on the curved path of the particle

Angular Velocity : As the linear velocity is defined as the time rate of change of displacement, angular velocity can be defined as the time rate of change of the angular displacement.

$\omega$ = $\frac{\Delta \theta}{\Delta t}$

where,
$\Delta \theta$ is the change in angle when the point is moving from one position P to other position Q.
$\Delta t$ is the change in time occurred for this angular displacement.

It can also be defined as the frequency of revolutions. $\omega$ = 2$\pi$ f

Angular Acceleration : As the linear acceleration is explained aw the time rate of change of velocity, angular acceleration is explained as the time rate of change of the angular velocity.

$\alpha$ = $\frac{\Delta \omega }{\Delta t}$

where,
$\Delta \omega $ is the change in angular velocity
$\Delta t$ is the change in time

Now,

$\Delta \omega$ = 2$\pi$ f

Therefore,

$\alpha$ = $\frac{2\pi f}{\Delta t}$

We have derived above that Tangential speed, $V_{t} = r\omega$
Therefore,

at= $\frac{d\left | v \right |}{dt}$

Now,

at = $\frac{d (r \omega)}{dt}$

Radius of curvature at a particular point is constant.

$a_{t}$ = $\frac{rd(\omega)}{dt}$

As explained in the explanation of angular acceleration, angular acceleration is equal to $\frac{d \omega }{dt}$

Therefore, Tangential acceleration is equal to the product of radius of curvature and the angular acceleration.
$a_{t}$ = r$\alpha$
where,
r is radius of curvature
$\alpha$ is the angular acceleration of the particle.

Tangential and Normal Components of Acceleration

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Acceleration at any point on the circular path is not always in the direction of the tangent t that particular point. That’s why the acceleration at any point on the circular path has two components. These two components are perpendicular to each other. One component is the tangential component of acceleration and the other component is the normal component of acceleration. Normal component of acceleration is also termed as the Radial component of acceleration.

Direction of the tangential component of acceleration at any particular point on the curved path is in the direction of the tangent at that particular point.

The radial component of acceleration is due to the centripetal force which is acting towards the center of the curved path. The resultant acceleration vector at any point on the curved path is the vector sum of the tangential vector and the radial component of the vector.

Angular and the Linear Acceleration Relation
:

We know that the linear acceleration can be defined as the rate of change of velocity and the angular acceleration is explained as the time rate of change of the angular velocity.

Relationship of angular and the linear acceleration can be expressed as the linear acceleration is explained as the multiplication of angular acceleration and the Radius of the circle.

Linear acceleration = Angular acceleration x Radius
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