Three masses (3 kg, 5 kg, and 7 kg) are located in the xy-plane at the origin, (2.3 m, 0), and (0, 1.5 m), respectively.
a) determine the center of mass of the system.
b) a 6 kg mass is added to the system. this four-mass system has a new center of mass at the origin. what is the location of the 6 kg mass?

a) C.M

b)

Explanation:

The center of mass "represent the unique point in an object or system which can be used to describe the system's response to external forces and torques"

The center of mass on a two dimensional plane is defined with the following formulas:

Where M represent the sum of all the masses on the system.

And the center of mass C.M

Part a

represent the masses.

represent the coordinates for the masses with the units on meters.

So we have everything in order to find the center of mass, if we begin with the x coordinate we have:

C.M

Part b

For this case we have an additional mass and we know that the resulting new center of mass it at the origin C.M and we want to find the location for this new particle. Let the coordinates for this new particle given by (a,b)

If we solve for a we got:

And solving for b we got:

So the coordinates for this new particle are:

be subject to a force andchange its motion