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Centrifugal Acceleration

The acceleration is the rate of change in the velocity of a moving body with respect to the given time. The acceleration can be classified according to the type of velocity and speed. If the body is in constant velocity then it is in the condition of constant acceleration because there is no change in velocity in the given time period. If a body is moved on circular path then it has a rotational acceleration due to rotational acceleration.

Similarly, some other types of accelerations are centripetal and centrifugal acceleration. Here we are discussing about the centrifugal acceleration. The centrifugal acceleration is the acceleration that is on the curved path. When a car takes a U turn on the curved way then it does not fall towards the centre. This is due to the centrifugal acceleration which makes the motion of the car. Let’s discuss the detail description of the centrifugal acceleration.

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What is Centrifugal Acceleration?

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Centripetal force is the force acting on the body which makes the body move towards the center of axis of rotation. This force acts along the radius towards the center of the circle.
There is another pseudo force acting opposite to that of centripetal force, which keeps the body move in a circular path with linear speed, away from the center of axis of rotation. This Force is called centrifugal force and if we consider its velocity we can see that the body moving in a circular path will be having linear velocity, so, such a change in the linear velocity taking place in this body with respect to time is called centrifugal acceleration.

Thus, centrifugal acceleration is the acceleration or change in velocity produced by the body moving in a circular path with respect to time, which keeps the body moving in a circular path without falling in to the center.

When an object is in motion along a curved path, it experiences a force and thereby an acceleration because the mass of the object remains constant. We will better explain taking a circular motion, as the concept is same. First, let us understand the concepts involved in a circular motion.

In case of circular motion, the velocity involved is called angular velocity.
An angular velocity is measured in terms of the angle covered by the object per unit time. Normally, an angular velocity is denoted by the Greek letter $\overline{\omega}$. The direction of angular velocity is limited to clock wise direction or counter clock wise direction.

Hence the definition of of average angular velocity is,

$\overline{\omega}$ = $\frac{\theta}{t}$, where, $\theta$ is the angle rotated in the time t.

Let us remember, although, the overall motion is circular, at any instant the object has a linear velocity v in a direction that is tangent to the circle of radius, say r, at that point.

Let $\iota$ be the actual distance moved by the object along the circumference. As per the geometry of circles,

$\theta$ = $\frac {\iota} {r}$ Therefore,
$\overline{\omega}$ = $\frac {\theta} {t}$
= $\frac {\iota}{(rt)}$
= $\left[(\frac {\iota}{t})(\frac{1} {r})\right]$,
or, $\overline{\omega}$ = $\frac {v} {t}$,
where, V = $\frac{\iota}{r}$ is the linear velocity of the object at any point.

Now, an angular acceleration is defined as the rate of change of angular velocity with respect to time. It is denoted by another Greek letter $\overline{\omega}$. Now considering an infinitesimal study,
$\overline{\omega}$ = $\frac {d\theta} {dt}$ and hence,
$\overline{\omega}$ = $\frac {d\omega}{dt}$, or $\overline{\omega}$ = $\frac {d^{2} \theta} {dt^{2}}$But an object is subjected to two types of acceleration in a circular motion.

Look at the following diagram.

Centrifugal Acceleration
ar = Direction of Centripetal Acceleration
at = Direction of Centrifugal Acceleration

Let's say an object is tied at A, one end of a string OA and rotated keeping the other end O as center. When the string is rotated fast, the string gets completely stressed out and the length of the string becomes the radius of rotation. It means a force is exerted on the object from the center to the end and there by an acceleration AO in a radial direction is faced by the object. This is called centrifugal acceleration. To counter this force a tension force developed in the string acting in the opposite direction. This tension force is called the centripetal force and the acceleration generated on the object is called the centripetal acceleration and denoted as ar. The opposite force which is acting tangentially is called centrifugal force which is denoted by at.

As per Newton’s third law, the magnitudes of centripetal and centrifugal forces and hence their acceleration components are equal.

Centrifugal Acceleration Equation

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For finding the centrifugal acceleration equation or a centrifugal acceleration formula, let us study the following diagram and derive the equation for centripetal acceleration and equate it to centrifugal acceleration.
Centrifugal Acceleration Equation
In figure (i), A and B are the positions of an object and the positions are infinitesimally close.
Figure (ii) shows the translated vector diagram of the centripetal velocity vectors at A and B. As per the similar triangles property,

$\frac{AB}{OA}$ = $\frac{\iota}{r}$ Since A and B are very close we can approximate AB, to the length of the arc AB and hence AB = v $\times$ dt
in fig (ii), since A and B are very close, Hence v + dv $\approx$ dv.

Therefore, $\frac{AB}{OA}$ = $\frac{dv}{v}$ becomes as,
$\frac {(v \times dt)}{(r)}$ = $\frac{dv}{v}$ or,
$(\frac {dv}{dt})$ = $\frac {(v^{2})}{(r)}$.Since $(\frac {dv} {dt})$ is the centripetal acceleration, we arrive at the formula as,
Centripetal Acceleration ar = $\frac {v^{2}} {r}$, and hence,

Centrifugal Acceleration at = $\frac {v^{2}}{r}$
(since centrifugal and centripetal forces are equal and opposite to each other).

It is expressed in m/s2.

The Effects of Centrifugal Acceleration

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The Centrifugal Accelerations due to centrifugal forces are perceived by us in everyday life. When you are traveling in a car and it takes a curve, you are pushed in a direction opposite to the radius of the curve. As long as this force of push is within a control you are safe because the inherent centripetal force takes care.

But as this force is directly proportional to the square of the linear speed of the car and inversely proportional to the radius of the curvature, there can be a danger of the car itself being thrown out radial if the linear velocity is high and the radius of curvature is small. 

This is the reason why the highways are banked wherever there is a curve on the high way and also why airplanes fly in an inclined positions when they take a turn. The idea behind such cases is, the horizontal component of the reaction force of earth on the car/airplane traveling/flying at the design velocity is more than the centrifugal force. 
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