Moments and Centroids
Mass and Slugs
Newton's Law states that
165 = 32m
In the metric system, kg is a mass unit and Newtons is a weight unit.
Moments and Center of Mass for Discrete Mass Points
Suppose that we have a teeter totter and a 10 kg child is on the left 5 meters from the center of the teeter totter and a 15 kg child is on the right 4 meters from the center of the teeter totter. We define the moment as:
10(-5) + 15(4) = 10
In general, we define the moment for masses mi at the points xi to be
Moment = S mi xi
center of mass = 10/25 = 0.4
We can say that if the center of the teeter totter was 0.4 meters from the current center, then the children would be in balance.
For points in the plane, we can find moments and centers of mass coordinate wise.
mx = (4)(-3) + (3)(2) + (1)(1) = -5
my = (4)(0) + (3)(2) + (1)(-2) = 8
Center of Mass = (-5/8,8/8) =
Center of Mass for a Two Dimensional Plate
First, we recall that for a region of density r bounded by f(x) and g(x)
Suppose that we revolve a region around the y-axis. Then the volume of revolution is:
V = 2prA
Suppose that we revolve the 4 x 4 frame with width 1 centered about (6,2) about the y-axis. Then we have that the Area is
A = 4 + 4 + 2 + 2 = 12
R = 7
V = 2p7(12) = 168p
(x - 11)2 + y2 = 4
about the y-axis.