Use integration to find the center of mass of the triangle (0,0) (2,0) (0,8).

Expert Answers

The center of mass of the the triangle with vertex (0, 0), (2, 0) and (0, 8) has to be determined.

The equation of the line passing through (0, 0) and (2, 0) is y = 0. The equation of the line passing through (0,0) and (0, 8) is x = 0 and the equation of the line passing through (2, 0) and (0, 8) is (y - 0)/(x - 2) = (8 - 0)/(0 - 2) = 8/-2 = -4

=> y = 8 - 4x

The diagram of the triangle is:

The center of mass of the triangle is given by:

`M_x = (int_0^2 (x*(8 - 4x))dx)/(int_0^2 8 - 4x dx)`

=> `(int_0^2 8x - 4x^2 dx)/(int_0^2 8 - 4x dx)`

=> `((8x^2/2 - 4x^3/3)_0^2)/((8x - 2x^2)_0^2)`

=> `((4x^2 - 4x^3/3)_0^2)/8`

=> `(16 - (4/3)*2^3)/8`

=> `2/3`

`M_y = (int_0^2 (1/2)(8 - 4x)^2 dx)/(int_0^2 8 - 4x dx)`

=> `(int_0^2 (1/2)(64 - 64x + 16x^2) dx)/(int_0^2 8 - 4x dx)`

=> `(int_0^2 32 - 32x + 8x^2 dx)/(int_0^2 8 - 4x dx)`

=> `(32x - 16x^2 + 8x^3/3)_0^2 / 8`

=> `8/3`

The center of mass of the triangle is `(2/3, 8/3)`

See eNotes Ad-Free

Start your 48-hour free trial to get access to more than 30,000 additional guides and more than 350,000 Homework Help questions answered by our experts.

Approved by eNotes Editorial