Question

Three heavy rods are all made of the same uniform material. These rods have lengths 3 m, 4 m, and 5 m, and compose the sides of a 3-4-5 right triangle. Find the coordinates of the center of mass of the triangle. (Assume that the right angle is at the origin, and that the two perpendicular sides are parallel to the axes, with the longer leg in the vertical direction.)

Answer 1

Answer:

[ 2.67 , 1 ] m

Explanation:

Given:-

- The side lengths of the rods are as follows:

                             a = 4 m , b = 4 m , c = 5 m

                             a = Base , b = Perpendicular , c = Hypotenuse

- All rods are made of same material with uniform density. With  

Find:-

Find the coordinates of the center of mass of the triangle.

Solution:-

- The center of mass of any triangle is at the intersection of its medians.

- So let’s say we have a triangle with vertices at points (0,0) , (a,0) , and (0,b).

• Median from (0,0) to midpoint (a/2,b/2) of opposite side has equation:

                                       bx−ay=0

• Median from (a,0) to midpoint (0,b/2) of opposite side has equation:

                                      bx+2ay=ab

• Median from (0,b) to midpoint (a/2,0) of opposite side has equation:

                                     2bx+ay=ab

• Solve all three equations simultaneously:

                                     bx−ay=0  , bx = ay

                                     ay + 2ay = ab , 3ay = ab , y = b/3

                                     bx = b/3

                                     x = a / 3

• So the distance from the median to each leg of the triangle is 1/3 length of other leg.

- So the coordinates of the centroid for right angle triangle would be:

                                   [ 2a/3 , b/3 ]

                                   [ 2.67 , 1 ] m