Centers of Mass

Suppose we have two particles of mass m1 and m2 at positions x1 and x2. If the particles are connected by a lightweight inflexible rod, then we say the particles form a system or a rigid body. In science and engineering applications, it is often useful to study properties of a system of particles. One of these properties is the center of mass for the system.
Figure 1
Roughly speaking, the center of mass is the point where the system would balance on the tip of a pin.

Group Discussion

Describe the center of mass for the two particle system if
  1. m1 = m2
  2. m1 > m2
  3. Predict where the center of mass is if m1 = 2*m2

The center of mass is important in understanding the dynamics of a system. For example, if we throw the "dumbell" shown in Figure 1, then the center of mass will follow a fairly simple path (roughly parabolic) whereas the ends will rotate around the center of mass, creating complicated paths as shown in Figure 2.

Figure 2

The center of mass of an object is the position at which the entire mass of the object could be concentrated, and still have the same "average motion" as the entire body does.

If a rigid body is composed of n particles connected in a straight line, then the location of the center of mass for the system is x_cm where
(m1 + m2 + . . . + m_n)x_cm = m1 x1 + . . . + m_n x_n
If the particles are extended into two or three dimensions the coordinates of the center of mass are defined by (x_cm, y_cm, z_cm) where
(m1 + . . . + m_n) x_cm = m1 x1 + . . . + m_n x_n
(m1 + . . . + m_n) y_cm = m1 y1 + . . . + m_n y_n
(m1 + . . . + m_n) z_cm = m1 z1 + . . . + m_n z_n

For a continuous distribution of matter, the summations turn into integrals. We want to think of the object as being cut into thin strips of small mass. For example, in Figure 3 the object is horizontally symmetric, so x_cm = 0.
Figure 3
To find y_cm, we can chop the object into strips of small mass: dm = p w(y) dy. Then y_cm satisfies the equation

Question 1

Find the center of mass of
  1. a rectangle
  2. a triangle
    Hint: Remember that y_cm is a constant.

Index

More: Centroids

Next: Moments of Inertia

Previous: Introduction

Jennifer Powell<jpowell@geom.umn.edu>
Fati Liamidi<liamidi@geom.umn.edu>