Question

Find the center of mass of a uniform parabolic sheet between y=a-x^2/a, y=0 and x=0

Answer 1

First we plot the graph. Now we take a smaller slice of length 'y' and thickness 'dx' along the x -axis.

The area of the slice is

the region is bounded by

The area of the bounded region is

  

The x coordinate of the center of gravity is

  

First we plot the graph. Now we take a smaller slice of length 'x' and thickness 'dy' along the y -axis.

The area of the slice is

The y coordinate of the center of gravity is

  

Let

So,

  

  

  

Since the parabolic sheet is uniform, center of gravity coinsides with center of mass.

So center of mass is