Centre of Mass of Isosceles Triangle?

[jereldc]

Homework Statement

an isosceles with base=a, height=h. It has uniform density and negligible thickness.

Homework Equations

determine it's centre of mass by integration,using more than one orientation

The Attempt at a Solution

It is a two-dimensional continuous structure. Let the base,a, lie on the X-axis and the Y-axis
be axis of symmetry. Then the COM must lie on the Y-axis. Slice the triangle into strips parallel to the X-axis. Each strip is of width dy, and length l. The mass per strip,dm, is obtained through:

mass=density x area,

p l dy
and the variable of integration is y as it is the axis of symmetry and integration is over 0 and h.

 

[LowlyPion]

Welcome to PF.

You will need to express l in terms y, since l is a function of y.

 

[nlightner]

Therefore, the total mass of the isosceles triangle is:

m=integral of (p l y) dy from 0 to h = p l h^2 / 2

The x-coordinate of the COM is given by:

xcom= integral of (x dm) from 0 to h / m = 0

since the density is uniform and x=0 at all points.

Similarly, the y-coordinate of the COM is given by:

ycom= integral of (y dm) from 0 to h / m = integral of (p l y^2) dy from 0 to h / p l h^2 / 2 = 2h/3

Therefore, the centre of mass of the isosceles triangle lies at (0,2h/3).