Center of Gravity?

Question

A ball with radius .060m and mass 1.00 kg is attached by a light rod .400m in length to a second ball with radius .080m and mass 4kg.

a.Where is the center of gravity of this system?
b.Suppose the rod is uniform and has a mass of 2kg. Where is the system's center of gravity now?

Answer 1

To find a center of gravity or centroid (coordinates) of a body you should compute the sum of the moments about a particular reference point ( a center of the coordinate system) and then divide that sum by the total mass (or area if density is uniform).
M=RxF (or in our case M=rm)
M-moment about a ref point
R, r - radius or distance from the center of gravity of mass m to a reference point

let the reference point be the center of the rod.
M1=(-.060- .200) 1= -.260 (' - ' sign since I'm assuming that it is located to the left of the reference point.

M2=(.200+ .080) 4= 1.12

Mt=M1+M2=-.260 + 1.12=0.86

r'= Mt/(m1+m2)=0.86/(1+4)=0.172m

Since it is symmetrical cal about the other axis

The center of gravity is located
R’= .06 + 0.200+ 0.172 m to the left from the center of the 1 kg ball or
R’=.080+.0200 – 0.172 m to the right from the center of the 4 kg ball

Answer 2

I am assuming that the balls are attached at the ends.

Method1:
Ignoring the radius of the balls.
0.4 meters length divided in half is 0.2 meters find the
distribution over 0.2 meters on either side of the center
of the rod. The distribution is the ratio of the kilograms.
4kg on left: 4/5=0.8=80%
1kg on the right: 1/5=0.2=20%

0.4*0.2=0.08 meters

Therefore, the ball with mass 4kg is on the left and the
center of gravity is 0.08 meters in from the left.


Method2:
f1d1=f2d2

d1=x
d2=0.4-x

1*x=4*(0.4-x)
1*x=1.6-4x
5*x=1.6
x=0.32 meters

Therefore, the 1 kg ball is 0.32 meters to the right of the
center of gravity and the 4 kg ball is 0.08 meters to the left
of the center of gravity.


Method3:

[(position1*weight1)+(position2*weight2)+...]/[(weight1)+
(weight2)+...]

[(0*1)+(0.4*4)]/[(1)+(4)]
=0.32 meters

Therefore, the 1 kg ball is 0.32 meters to the right and the
4 kg ball is 0.08 meters to the left.

Personally, I think method number 2 makes the most
sense. The center of gravity is where the entirety of the
weight will be balanced on a fulcrum. The question
becomes where to place the fulcrum to make the
plank balance.

b)
The center of gravity will not change due to the uniform
distribution of 2 kg or any mass for that matter.