A 30.0 g mass is located at the origin. Where must a mass of 10.0 g be located if the coordinates of the center of mass are (0.00 cm, 10.0 cm)?

The center of mass (COM) of a system of particles is given by the formula:

COM = (m1r1 + m2r2) / (m1 + m2)

where m1 and m2 are the masses of the particles and r1 and r2 are their respective positions.

In this case, we have:

m1 = 30.0 g (mass at the origin) r1 = 0.00 cm (position of the mass at the origin)

m2 = 10.0 g (mass we are trying to find the position of) r2 = ? (this is what we're trying to find)

The coordinates of the center of mass are given as (0.00 cm, 10.0 cm). Since we're only dealing with one dimension here, we can ignore the x-coordinate and just use the y-coordinate of 10.0 cm.

So, we can set up our equation as follows:

10.0 cm = (30.0 g * 0.00 cm + 10.0 g * r2) / (30.0 g + 10.0 g)

Solving for r2 gives:

r2 = 10.0 cm * (30.0 g + 10.0 g) / 10.0 g r2 = 40.0 cm

So, the 10.0 g mass must be located at 40.0 cm from the origin.