Circular motion

In the diagram v is the tangential velocity of the object. a is the centripetal (acting towards the center of the circle) acceleration and Fis the centripetal force. r is the radius of the circle and m is mass of the object.

Center of mass

Consider that there are a number of particles forming a system, such that, when an external force is applied, the complete system moves as on. Now, if we are able to replace all these particles by just one particle. And, this new particle is placed in such a manner and has such a mass, that when acted on by an external force, it moves exactly in the same manner as the system it replaced. Then, we have found the center of mass of the system. (This sounds more complex than it is. So don't worry! read on)

Center of mass of a two particle system

Let us now consider two particles. Let them have masses m1 and m2. Let r1(t) and r2(t) be the position vectors of the two particles at the instant of time t. (Note: On a web page it is difficult to use the usual vector notation. So I will use bold fonts to depict vectors). The position vector Rc.m.(t), giving the center of mass of this system is given by the formula.

Rc.m.(t) = {m1r1(t) + m2r2(t)} / (m1 + m2)

from the above it is obvious that for a two particle system the center of mass always lies between the two particles. It also lies on the line joining the two particles.

Center of mass of a large number of particles

Rc.m.(t) = {m1r1(t) + m2r2(t) + ----- mn rn(t)} / (m1 + m2 + ------ mn)

Center of mass of a rigid body

The position of the center of mass of a rigid body is a fixed point. This point may or may not lie in the body. Where is lies depends upon the shape of the body. For example of we take the case of a ring. The center of mass will lie at its geometrical center. Obviously this point does not lie in the ring.

Center of mass of some rigid bodies

Body  Center of mass    

Rod

Mid point of rod

Ring

Center of ring

Sphere

Center of sphere

Angular velocity

Average angular velocity = Angular displacement / time interval

So if ω is the average angular velocity, θ the angular displacement and t the time interval. We can say

 

So we can write the instantaneous angular velocity as dθ /dt

Linear velocity

Linear velocity = Radius of the circular path x Angular velocity

 

Angular acceleration

Average angular acceleration = Change in angular acceleration / Time interval

Instantaneous angular acceleration

Linear acceleration

Linear acceleration = Radius of the circular path x Angular acceleration

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