In this article we are going to illustrate some facts on angular acceleration and centripetal acceleration in a detailed manner.
Before starting to describe the facts related to the angular acceleration and centripetal acceleration we should have the basic idea of these two accelerations. The rate of change of angular velocity of a spinning body around a fixed axis can be defined as angular acceleration of that spinning body. In case of centripetal acceleration of a moving body in a circular path,the direction of velocity changes continuously.
When a body rotates around a fixed axis it moves with a constant speed. As this body keeps rotating,its direction along with its velocity changes. Due to this change in velocity angular acceleration is produced which is also termed as rotational acceleration. Its unit is radian/second^2. Angular acceleration(⍺) can be written as,
⍺ = dw/dt
Where w is the angular velocity.
A constant speed is always tried to be retained by a moving body in a circular path,but that does not mean it moves with a constant velocity. As its direction changes from time to time the velocity(v) vector also changes from time to time. This rate of change of velocity(v) is known as the centripetal acceleration. It is always directed towards the center. The expression for the centripetal acceleration(a) is,
Where r is the radius of the circular path.
Is angular acceleration and centripetal acceleration the same?
No,angular acceleration and centripetal acceleration are not the same. They are two different quantities. Though they have a similarity, they are not the same.
The similarity is that angular acceleration and centripetal acceleration both are vector quantities.
The differences or dissimilarities between these angular acceleration and centripetal acceleration are stated below:
- They have a difference in their units. The unit of angular acceleration is radian/second2 and the unit of centripetal acceleration is meter/second2.
- Angular acceleration is related to the angular motion of a rotating body around a fixed axis whereas centripetal acceleration is related to the motion of a body in a circular path.
- Angular acceleration has a fixed direction which is the direction of the axis i.e,the direction of the corkscrew rule whereas the centripetal acceleration is directed towards the center of the circular path which changes from time to time.
- Angular acceleration is a derived quantity related to the angular motion of a body but centripetal acceleration is a derived linear quantity.
- Whenever an object is rotating around a fixed axis with a constant angular velocity(w) its angular acceleration is zero but centripetal acceleration gains a finite value as,centripetal acceleration a= w2r where r is the radius of the path.
Angular acceleration and centripetal acceleration same relation
No, the expression or relation for the angular acceleration and centripetal acceleration is not the same.
We know that angular velocity(dw)= angular displacement/time
ϴ1 is the position of the particle moving around a fixed axis at time 0 second and ϴ2 is the position of the particle at time t seconds. Here dϴ=(ϴ2-ϴ1),dt=(t-0)
Therefore angular acceleration is,⍺ = dw/dt
⍺ = d2ϴ/dt2
And hence their units are also different. The unit of angular acceleration is rad/second^2 and the unit of centripetal acceleration is m/s2.
Does angular acceleration affect centripetal acceleration?
No, the angular acceleration does not affect the value of the centripetal acceleration of a body moving in a circular path but it affects the total linear acceleration of that body.
We know that the value of centripetal acceleration(ac) is, ac = vt 2/r. Where vt is the tangential linear velocity which is equal to wr( w is the angular velocity and r is the radius of that circular path).
The reason due to which centripetal acceleration is produced is the change in the direction of the tangential velocity and the change in its magnitude is the reason behind the production of tangential acceleration.
The tangential acceleration and the centripetal acceleration are always normal to each other. The magnitude of the total linear acceleration(a) vector of a rotating rigid body around the circular path of radius r is,
a2 = ac 2 + at 2
For uniform circular motion of a body the angular acceleration remains zero. Hence the total linear acceleration depends upon the centripetal acceleration.
Difference between centripetal acceleration and angular acceleration
The differences that have been discussed earlier were regarding the mathematical expression,units etc of these two accelerations. But now we will differentiate these two accelerations by their corresponding examples.
Examples of centripetal acceleration are stated below:
A very common example is the movement of a stone that is tied to one end of a string. Now if we apply the centripetal force to rotate the string it will be seen that the stone is moving in a circular path and the direction of the centripetal acceleration is towards the center.
Another example is the movement of the earth around the sun. Here the required centripetal force for this movement is the gravitational force.
Examples of angular acceleration is stated below:
The movement of the blades of a fan is the example of angular acceleration.
Another example is the wheel and axle arrangement or the rotation of the wheel of a cycle around the fixed axis.
How to find angular acceleration from centripetal acceleration?
The formula for the centripetal acceleration is,
Now we all know that linear velocity v and angular velocity w are related to each other by the relation v= wr where r is the radius of the path. From this related it can be stated that,a= (w.r)2/r
If the values of the radius and the centripetal acceleration are available then we can calculate angular velocity from them. We know that angular acceleration is, ⍺ = dw/dt.
Hence from the calculated value of angular velocity and the value of time that is provided in the question we can calculate the value of angular acceleration.
In this article we have discussed the relation between angular acceleration and centripetal acceleration,how we can find angular acceleration from the centripetal acceleration,the dependence of angular acceleration over centripetal acceleration etc.