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.

Related Calculators | |

Vector Magnitude Calculator | Acceleration Calculator |

Acceleration due to Gravity Calculator | Angular Acceleration Calculator |

a =

= $\frac{\Delta{v}}{\Delta{t}}$.

In terms of dimensions, we can define it as,

=

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

and

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

Thus, Dimensional Formula is

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,

r

If we Consider free fall of body, the magnitude of acceleration is conventionally denoted by **g** and has approximate value of **9.8m/s**^{2}.

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.

In case of circular motion the velocity involved is called angular velocity.

Hence the definition of of average angular velocity is,

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

where

Now, we will bring in the linear velocity

Let

$\theta$ = $\frac{\iota}{r}$

Therefore,

$\omega$ = $\frac{\theta}{t}$ = $\frac{\iota}{(rt)}$

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

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

or,

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

where

Now,

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

and hence,

$\alpha$ = $\frac{d\omega }{dt}$

or

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

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

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

The situation is described in the following diagram

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

Related Topics | |

Physics Help | Physics Tutor |