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:

X_com = (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.

In some situations it's necessary to integrate to find the center of mass. The integral equation looks like:

X_com = ∫ x dm / M

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:

X_com = (x₁m₁ + x₂m₂ + x₃m₃) / (m₁ + m₂+ m₃ )

X_com = (0.4 x 1.2 + (-0.4) x 2.1 + 0.1 x 1.7) / (1.2 + 2.1 + 1.7)

X_com = -0.038

Y_com = (y₁m₁ + y₂m₂ + y₃m₃) / (m₁ + m₂+ m₃ )

Y_com = (0.7 x 1.2 + 0.7 x 2.1 + 0.2 x 1.7) / (1.2 + 2.1 + 1.7)

Y_com = 0.53