Consider a square ABCD of side 10cm and total mass 10kg. Let O be the center of the sqaure. We cut a square of corners O and C.
Find the central mass of the obtained shape after removing the small cutten sqaure.
To find the central mass of the obtained shape after removing the small cut square, we need to determine the position of the center of mass for each of the remaining parts.
1. Square ABCD: The center of mass of a square lies at its geometric center. Since the square has equal sides, the center of mass lies at point O, which is also the center of the square.
2. Cut square: The cut square has two triangular pieces left (triangle AOC and triangle ODC). Each of these triangles has the same shape and mass, so their center of mass will be located at the midpoint of the line segment connecting their common vertex (O) to the midpoint of the opposite side.
To find the position of the center of mass of the triangles, consider the following:
- The length of the side of the cut square is 10 cm, so the distance from O (the vertex) to the midpoint of one side is 5 cm.
- The distance from O to the midpoint of the opposite side is equal to the diagonal length divided by 2. The diagonal length is found using the Pythagorean theorem:
Diagonal length = √(10^2 + 10^2) = √200 = 10√2 cm
- Therefore, the distance from O to the midpoint of the opposite side is (10√2)/2 = 5√2 cm.
- Since the diagonal length divides the triangle into two congruent right triangles, the center of mass for each triangle will lie along the median (line connecting the midpoint of the hypotenuse to the opposite vertex).
- Using similar triangles, we find that the ratio of the distance from O to the center of mass to the distance from the midpoint of the hypotenuse to the opposite vertex is 1:2. Therefore, the distance from O to the center of mass of each triangle is (5/3)√2 cm.
Now, we can calculate the combined position of the center of mass for the remaining parts by taking the weighted average:
- The square ABCD has a mass of 10 kg, and its center of mass is located at O.
- The cut square has a mass of (10/3) kg (since it forms a quarter of the original square ABCD), and its center of mass is located at the midpoint between the centers of mass of the two triangles.
- The distance between the center of mass of the triangles and point O is (5/3)√2 cm (as calculated earlier).
Using these values, we can calculate the position of the combined center of mass:
x-coordinate of center of mass =
[(mass of square × x-coordinate of O) + (mass of cut square × x-coordinate of center of mass of triangles)] / (mass of square + mass of cut square)
y-coordinate of center of mass =
[(mass of square × y-coordinate of O) + (mass of cut square × y-coordinate of center of mass of triangles)] / (mass of square + mass of cut square)
Plug in the respective values and perform the calculations to find the x-coordinate and y-coordinate of the center of mass of the obtained shape after removing the small cut square.