where the density at the point (x,y) is proportional to the distance between
(x,y) and the x axis.
1. Find the center of mass of the lamina corresponding to the parabolic region
where the density at the point (x,y) is proportional to the distance between
(x,y) and the x axis.
Solution:
Because the lamina is symmetric with respect to the y axis:
the center of mass lies on the y axis. Thus xbar=0. To find ybar, first find
the mass of the lamina (we treat k like a constant):
Next, we must find the moment about the x axis:
Thus:
and the center of mass is
2. Find the moment of inertia about the x-axis of the lamina in the above
example.
Answer:
From the definition of moment of inertia, you have:
