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Magnitude of Acceleration

Acceleration means the change of velocity of an object with respect to time. The ratio of change in velocity to the ratio of change in time in a given interval is the average acceleration. in that time interval. In simple words we can define acceleration, it means speeding up. The S. I. unit of acceleration is metre per second square and can be written as ms^{-2}. It is a vector Quantity.

We know that magnitude is something where only distance covered is considered but not the displacement.

Generally, the motions of objects are classified as motion in a straight line in a given time interval or the motion may be along a curve. The most common motion along a curve is the circular motion or rotation. 
The accelerations are broadly of two types :
  • Linear acceleration in case of linear motion
  • Angular acceleration in case of circular motion
The magnitude of acceleration is something where we are considering only magnitude but not the direction. Of course, there will be change in direction in velocity in acceleration but we are not going to deal with that since we are considering only the circular motion, where direction is limited to two. In case of linear motions, the direction may be anything but in case of circular motion the directions are limited to two. The object may move in a clockwise direction or in a counter clock wise direction around the center of motion. The latter is considered positive as per convention.

In this article, let us concentrate on circular motion and hence the angular acceleration. We will consider only the counter clock wise rotation and will focus only on magnitude of angular acceleration. 

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Formula

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Defined as Increase in Velocity to the corresponding short interval of time.Here,

a =
$\frac{Increase\ in\ Velocity}{Short\ Time\ Interval}$
   
   = $\frac{\Delta{v}}{\Delta{t}}$.

In terms of dimensions, we can define it as,
a = $\frac{L}{T}$ $\times$ $\frac{1}{T}$
 
   = LT -2
where,

$\Delta$ v = $\frac{L}{T}$

and

$\Delta$ t = $\frac{1}{T}$.

Thus, Dimensional Formula is [LT -2].

In uniform circular motion, the direction of acceleration is towards the center of the circular path that is called centripetal acceleration and the constant of magnitude of acceleration a is given by,
a = $\frac{v^{2}}{r}$.
Where, v denotes the speed in m/s, and

r
is the radius of circular path in m.

If we Consider free fall of body, the magnitude of acceleration is conventionally denoted by g and has approximate value of 9.8m/s2.

Magnitude of Angular Acceleration

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Before proceeding to the magnitude of angular acceleration and deriving a formula for the magnitude of angular acceleration, let us refresh the fundamentals of the circular motion.

In case of circular motion the velocity involved is called angular velocity. Angular velocity/rotational acceleration is measured in terms of the angle covered by the object per unit time. Normally, an Angular velocity is denoted by the Greek letter $\omega$. The direction of angular velocity is just two, limited to clockwise direction or counter clockwise direction. 

Hence the definition of of average angular velocity is,
$\omega$ = $\frac{\theta }{t}$,

where $\theta$ is the angle rotated in the time t.

Now, we will bring in the linear velocity v and the radius of the circle r, in a circular motion.

Let $\iota$ be the actual distance moved by the object along the circumference. As per the geometry of circles, we will bring in the linear velocity v and the radius of the circle r, in a circular motion.
$\theta$ = $\frac{\iota}{r}$

Therefore,
$\omega$ = $\frac{\theta}{t}$ = $\frac{\iota}{(rt)}$
= $\left [\left(\frac{\iota}{t} \right ) \left ( \frac{1}{r} \right )\right ]$,

or,
$\omega$ =$\frac{v}{r}$,

where v 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 $\alpha$. Now considering an infinitesimal study,

$\omega$ = $\frac{\mathrm{d}\theta}{\mathrm{d} t}$

and hence,
$\alpha$ = $\frac{d\omega }{dt}$

or 

$\alpha$ = $\frac{d^2\theta }{dt^2}$.

$\alpha$ = $\frac{\frac{v}{r}}{\frac{1}{r}}$

[$\because$$\frac{\mathrm{d} \omega}{\mathrm{d} t}$ = $\frac{v}{r}$$\times$$\frac{1}{r}$].

$\alpha$ = $\frac{v^{2}}{r}$.

But, in a circular motion two types of accelerations exist. Let us explain this concept with a simple experiment.

Take a string and securely tie an object at one end, hold the other end and start rotating the string. You will notice that the string gets completely stretched. 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. 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 ac. Suppose the string snaps right at the end where the object is tied. The object, now, will invariable fly in a direction tangent to the circle at the point. This is because there exists a tangential force on the object and the acceleration due to this force is called the tangential acceleration and denoted as at.
The situation is described in the following diagram

Angular Acceleration

The string is turned in clockwise or anticlockwise direction, the magnitude of acceleration will be the same, which is given by,a = $\frac{v^2}{r}$.
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