Question

Write the expression for position vector of centre of mass of a rigid body and explain the terms ?
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Answer 1

Answer:

Write the expression for position vector of centre of mass of a rigid body and explain the terms

Answer 2

To find:

The expression for position vector of centre of mass of a rigid body ?

Solution :

Let us consider a rigid body of total mass m.

Now, in order to calculate the position of centre of mass, we need to imagine the body to be composed of multiple small components located at varying positions from the origin.

So, the x axis component of position will be :

$\therefore \: \bar{x} = \dfrac{ \sum( m_{i} x_{i} )}{ \sum( m_{i})}$

$\boxed{ \implies\: \bar{x} = \dfrac{m_{1} x_{1} + m_{2} x_{2} + .....n \: times}{m_{1}+m_{2} + .....n \: times } }$

Similarly , the y-axis component of position will be:

$\therefore \: \bar{y} = \dfrac{ \sum( m_{i} y_{i} )}{ \sum( m_{i})}$

$\boxed{ \implies\: \bar{y} = \dfrac{m_{1} y_{1} + m_{2} y_{2} + .....n \: times}{m_{1}+m_{2} + .....n \: times } }$

Similarly , the z axis component will be :

$\therefore \: \bar{z} = \dfrac{ \sum( m_{i} z_{i} )}{ \sum( m_{i})}$

$\boxed{ \implies\: \bar{y} = \dfrac{m_{1} z_{1} + m_{2} z_{2} + .....n \: times}{m_{1}+m_{2} + .....n \: times } }$

• Kindly note that we have assumed the rigid body to be composed of "n" number of particles each having their own discrete mass values.

Hence , the final coordinate will be expressed as :

$\boxed{ \therefore \vec{ (COM)} = \bar{x} \hat{i} + \bar{y} \hat{j} + \bar{z} \hat{k}}$