When an object is thrown into the air, different parts of the object can follow complicated paths if the object spins as it travels. However, the center-of-mass of the object will always follow a parabolic trajectory through the air.

The center-of-mass is the point that moves as though all the mass is concentrated there.

The center-of-mass of an object, or a collection of objects, can be found using:

Xcom

=

x1m1 +x2m2 + ... m1 + m2 + ...

That tells you the x-coordinate of the center of mass. The y-coordinate and z-coordinate can be found from equivalent expressions.

If you have mass distributed in some way over an object instead of having a number of discrete masses, it's generally necessary to integrate to find the center of mass. The integral equation looks like:

Xcom

=

∫ x dm ∫ dm

=

∫ x dm M

where M is the total mass of the object

The center-of-gravity is the point that moves as though all the weight is concentrated there.

In a uniform gravitational field, weight and mass are proportional to each other and the center-of-mass and the center-of-gravity are the same point.

They're different if the gravitational field is non-uniform. We're not going to worry about that.

The center-of-gravity can be determined by hanging an object from a support. At equilibrium, the center-of-gravity will always hang directly below the support.

Three balls have masses and positions as follows:

• Ball 1: m₁ = 1.2 kg, x₁ = 0.4 m, y₁ = 0.7 m.
• Ball 2: m₂ = 2.1 kg, x₂ = -0.4 m, y₂ = 0.7 m.
• Ball 3: m₃ = 1.7 kg, x₃ = 0.1 m, y₃ = 0.2 m.

The center-of-mass of this system is given by:

Xcom

=

x1m1 + x2m2 + x3m3 m1 + m2 + m3

Xcom

=

0.4 * 1.2 + (-0.4) * 2.1 + 0.1 * 1.7 1.2 + 2.1 + 1.7

X_com = -0.038 m

Ycom

=

y1m1 + y2m2 + y3m3 m1 + m2 + m3

Ycom

=

0.7 * 1.2 + 0.7 * 2.1 + 0.2 * 1.7 1.2 + 2.1 + 1.7

Y_com = 0.53 m