Centroids

The centroid corresponds to the geometric center of an object. If the material is uniform, the centroid is where the figure would balance on your finger.

Question 1

Solve for the centroid of the T-beam and the C-beam

Note that the T-beam is composed of 2 pieces
and the C-beam is composed of 3 pieces

Question 2

  1. Compute the centroid of each piece
    T1=
    T2=
    C1=
    C2=
    C3=
  2. Determine how the centroid of a composite structure relates to the centroids of the component pieces.
  3. What happens to the location of the centroid of the T-beam if we instead decompose the beam as
    and the C-beam as
  4. Based on your observations above, formulate a conjecture that relates the position of the centroid of a beam to the location of the component parts.
  5. One old Victorian home has ornate carved beams like the one below.

    How might you determine the centroid of an irregularly shaped region such as this?

Question 3

Suppose you are given a composite region in which the density varies across the regions (but is constant within each region) as shown below:

  1. Use the properties of integrals to find an expression for the centroid of the composite region, given the centroids and masses of the individual sections.
  2. Can you generalize to a region composed of arbitrarily many (symmetric) sections?

Index

Next: Moments of Inertia

Previous: Introduction

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