Consider two particles located along the x-axis as shown.

The mass of particle A is 5 grams, and the mass of particle B is 20 grams.

The center of mass of the two-particle system is the x value at which if we placed a fulcrum, the two masses would be balanced.

The center of mass is given by the sum of the products of the position and mass of each particle, all divided by the total mass of the system.

Consider a baseball bat that is one meter long. It is made from wood with a constant density.

We can write a generalized form of the definition in order to calculate the center of mass for n particles.

The bat's shape can be approximated by rotating the given function about the x-axis.

Knowing the volume of the bat, we can calculate its mass.

Now, suppose we wish to balance the bat on a fulcrum.

Use the slider to vary the position of the fulcrum, and find the center of mass of the bat.

We can also calculate the center of mass. However, in this case we do not have a system of a few particles with various masses.

Instead, we think of the bat as a collection of an infinite number of particles.

Each particle has a mass given by the product of its density and volume.

We can rewrite the center of mass equation in terms of a definite integral and the total mass.

Now, consider a rigid, horizontal rod of length L.

The linear density of the rod is not uniform, but varies as a function of x.

The total mass of the rod is obtained by integrating the density function over the length of the rod.

Suppose we have a rod that is four meters long, with the given density function.

The center of mass of a non-uniform rod is given by the definite integral of x times the linear density function, which is then divided by the total mass.

Notice that if the rod has a uniform linear density k, then the center of mass is at the center of the rod, as we would expect.

What is the center of mass of this two-particle system? Click the "Submit" button after selecting your answer.

What is the volume of the bat? Click the "Submit" button after selecting your answer.

What is the center of mass of the baseball bat? Click the "Submit" button after selecting your answer.

What is the total mass of the rod? Click the "Submit" button after selecting your answer.

What is the center of mass of our rod? Click the "Submit" button after selecting your answer.

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