Question

If there are two particles of masses m1 and m2 have coordinates x1 and x2. What is the expression for coordinate of center of mass of that two particle system?

Answer 1

Given : two particles of masses m₁ and m₂ have coordinates x₁ and x₂.

To Find : expression for coordinate of center of mass of that two particle system

Solution:

expression for coordinate of center of mass of that two particle system

= (m₁x₁ + m₂ x₂)/(m₁ + m₂)

the center of mass of a distribution of mass in space is the unique point    where the weighted relative position of the distributed mass adds to zero.

For simple rigid objects with uniform density,  center of mass is located at the centroid of object.

(m₁x₁ + m₂ x₂)/(m₁ + m₂)

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Answer 2

Given:

There are two particles of masses m1 and m2 have coordinates x1 and x2 respectively.

To find:

Expression for coordinate of center of mass of that two particle system?

Solution:

The coordinates can be taken as (x1,0) and (x2,0):

The expression for the centre of mass can be given as :

X axis coordinate:

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

$\sf \implies\: \bar{x} = \dfrac{m_{1}x_{1} + m_{2}x_{2} }{m_{1} + m_{2}}$

Y axis coordinate:

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

$\sf \implies\: \bar{y} = \dfrac{m_{1}y_{1} + m_{2}y_{2} }{m_{1} + m_{2}}$

$\sf \implies\: \bar{y} = \dfrac{m_{1}(0)+ m_{2}(0)}{m_{1} + m_{2}}$

$\sf \implies\: \bar{y} = \dfrac{0}{m_{1} + m_{2}}$

$\sf \implies\: \bar{y} = 0$

So, coordinates of COM:

$\boxed{ \sf \: (\bar{x}, \bar{y} )= \bigg(\dfrac{m_{1}x_{1} + m_{2}x_{2} }{m_{1} + m_{2}},0 \bigg)}$