Mae 241 Lec14

9. Center of gravity and centroid

• Discuss the concept of center of gravity, center of mass and centroid • Show how to determine location of the center of gravity of bodies

9.1 Center of gravity, center of mass and centroid of a body Center of gravity • A body is composed of an infinite number of particles with a weight dW • These weights are forming a parallel force system. The resultant of this force system is the total weight of the body • The total weight W passes through a single point called the center of gravity G

9.1 Center of gravity, center of mass and centroid of a body Center of gravity ~

−

x=

∫ x dW ∫ dW

~

y dW ∫ y= ∫ dW −

~

z dW ∫ z= ∫ dW −

Coordinates of the center of gravity G Coordinates of each particle in the body

9.1 Center of gravity, center of mass and centroid of a body Center of mass of a body • In order to study the dynamic response or accelerated motion of a body is important to locate the body’s center of mass Cm • Substitute dW = g dm into the previous equations (g is constant and cancels out) ~

~

x dm ∫ x= ∫ dm

y dm ∫ y= ∫ dm

−

−

~

z dm ∫ z= ∫ dm −

9.1 Center of gravity, center of mass and centroid of a body Centroid of a volume • If the body is made from a homogeneous material its density ρ will be constant • Therefore, a differential element of volume dV Has a mass dm = ρ dV • Substituting dm into the previous equations (ρ cancels out) ~

−

~

∫ xdV

−

x=V

y=V

∫ dV

∫ dV

V

~

−

∫ ydV

∫ zdV

z=V

∫ dV

V

V

9.1 Center of gravity, center of mass and centroid of a body Centroid of an area • The area lies in the x-yplane and is bounded by the curve y = f(x) • The centroid can be determined from from the integrals below ~

−

x=

∫ xdA A

∫ dA A

~

−

y=

∫ ydA A

∫ dA A

9.1 Center of gravity, center of mass and centroid of a body Centroid of a line • The area lies in the x-yplane and is bounded by the curve y = f(x) • The centroid can be determined from from the integrals below ~

−

x=

∫ xdL L

∫ dL L

~

−

y=

∫ ydL L

∫ dL L

dL can be determined from Pythagorean theorem

Important points

• Centroid represents the geometric center of the body. • This point coincides with the center of mass or center of gravity ONLY if the material composing the body is uniform or homogeneous