Why is an object perfectly balanced only at the center of mass?

Question

What makes it so? This is not homework. Thanks.

Answer 1

Let's take a simple example, and imagine a lever on a fulcrum. Let's imagine that the fulcrum is right at the center, so that the lever is balance. Now, we can put 2 different weights on this beam, m1 and m2, on opposite sides of the fulcrum. The torque (twisting force, causing the beam to rotate) exerted on the beam by each of the weight is calculated as follows:

T = m d

where T is the torque, m is the mass, and d is the distance the mass is from the center of rotation, or where the fulcrum is. In order for the torque of both masses m1 and m2 to balance out, we must have this condition satisfied:

m1 d1 = m2 d2

which is the same as saying that both masses exert the same but opposite torque on the lever, so that the lever does not turn. This happens to define where the center of gravity is for the two masses m1 and m2. If you moved the fulcrum anywhere else, the torques exerted by the masses won't balance out.

Answer 2

Because that's the definition of center of mass (COM).

COM is the point where, if you put a plane through the object (any plane), half the mass is on one side, and half on the other.

Or in reverse: imagine cutting the object in half, so that 1/2 the mass is in each half. Now cut it in a different direction, again so that the mass is half and half. Where the 2 cuts meet is a line.
Now make a third cut, in a new direction; still half & half. The 3rd cut will meet the line at some point. That point is the COM.

It will always be the same point, no matter how you make the cuts (as long as the cuts are in 3 different planes).

Answer 3

At the centre of mass there are no moments about that point. Since at that point the clockwise moment=anticlockwise moment. W1D1=W2D2