3 PARTICLES OF MASS 1KG,2KG,3KG PLACED AT CORNER A,B,C OF AN EQUILARERA TRIANGLE A,B AND C.LOCATE THE POSITION OF PARTICLE/FIND COORDINATES
CM of the system = (7/12, √3/4)
3/12,√3/4
To find the position or coordinates of the particles in the equilateral triangle, we can use some basic trigonometry.
Let's assume that the triangle ABC is equilateral, with each side length denoted as 's'.
The center of the equilateral triangle is the centroid, which is equidistant from each vertex. Let's call the centroid 'O'.
Now, if we know the side length (s) of the triangle, we can determine the coordinates of the vertices A, B, and C.
Assume that vertex A is located at the origin (0, 0).
In an equilateral triangle, the coordinates of vertex B are given by (s, 0), and the coordinates of vertex C can be found using the Pythagorean theorem.
The distance from A to C is equal to [s / 2 * sqrt(3)]. So, if we move along the y-axis from A, the coordinates of vertex C will be (0, s / 2 * sqrt(3)).
Now we know the coordinates of the three vertices: A (0, 0), B (s, 0), and C (0, s / 2 * sqrt(3)).
To determine the position of each particle, we need to find their coordinates relative to the centroid (O), which is the center of mass of the equilateral triangle.
The centroid O divides each median of the triangle into two equal parts. The distance from each vertex to the centroid is equal to (2/3) of the median length.
So, the distance from vertex A to the centroid O is (2/3) * s / 2 * sqrt(3) which can be simplified as (1/3) * s * sqrt(3). This is the y-distance.
The distance from vertex B to O is also (1/3) * s * sqrt(3), but the x-coordinate of O is s / 2.
Similarly, the distance from vertex C to O is (1/3) * s * sqrt(3) and the coordinates of O are (0, s / 6 * sqrt(3)).
Now, we can find the coordinates of each particle relative to the centroid O by using the x and y coordinate differences.
For particle 1 of mass 1kg, its coordinates are (0 + x1, 0 + y1), where x1 and y1 are the x and y coordinate differences between O and A, respectively.
For particle 2 of mass 2kg, its coordinates are (s / 2 + x2, 0 + y2), where x2 and y2 are the x and y coordinate differences between O and B, respectively.
For particle 3 of mass 3kg, its coordinates are (0 + x3, s / 2 * sqrt(3) + y3), where x3 and y3 are the x and y coordinate differences between O and C, respectively.
These coordinate differences can be found by subtracting the x-coordinate of O from the x-coordinate of the respective vertex and subtracting the y-coordinate of O from the y-coordinate of the respective vertex.
Finally, the coordinates of each particle can be expressed as follows:
Particle 1: (x1, y1)
Particle 2: (s / 2 + x2, y2)
Particle 3: (x3, s / 2 * sqrt(3) + y3)
Please note that the actual values of x1, y1, x2, y2, x3, and y3 will depend on the specific side length 's' of the equilateral triangle.