Video Transcript
In a rectangle 𝐴𝐵𝐶𝐷, 𝐴𝐵 equals 22 centimeters and 𝐵𝐶 equals 26 centimeters. Four masses of six, seven, five, and nine grams are placed at the vertices 𝐴, 𝐷, 𝐵, and 𝐶, respectively. Another mass of magnitude eight grams is attached to the midpoint of line segment 𝐴𝐷. Determine the coordinates of the center of mass of the system given that 𝐶 is placed at the origin and the scales of the axes are defined such that each unit represents one centimeter of distance.
The diagram below represents the given information with the values in the circles representing the masses at points 𝐴, 𝐵, 𝐶, and 𝐷. In addition, we have a mass of eight grams attached at the midpoint of line segment 𝐴𝐷. In order to work out the coordinates of the center of mass of this system of masses, we’ll need to work out the coordinates and the position vectors of each mass. We can use the given length of the two sides of the rectangle to help us do this.
We are given that 𝐴𝐵 is 22 centimeters and 𝐵𝐶 is 26 centimeters. Of course, because it’s a rectangle, we know that opposite sides are congruent. So, 𝐴𝐷 is also 26 centimeters and 𝐷𝐶 is 22 centimeters. Because 𝐶 is placed at the origin, we know that its coordinates will be at zero, zero. As a position vector, we could give this as zero 𝑖 plus zero 𝑗.
Next, we can look at the coordinates of 𝐵. It’s 26 centimeters along the 𝑥-axis. And because each unit on the coordinate grid represents one centimeters, we can say that this is at the coordinates 26, zero. As a position vector, we could write this as 26𝑖 plus zero 𝑗. For point 𝐴, the coordinates can be given by moving 26 along and 22 up. So that’s 26𝑖 plus 22𝑗 as a position vector. The position vector of 𝐷 can be given as zero 𝑖 plus 22𝑗 because that would be at the coordinate zero, 22. We then need to work out the position vector of this final mass of eight grams, which we’re told is at the midpoint of line segment 𝐴𝐷. If we wanted to travel to this point, we would move 13 to the right and 22 up. As a vector, we would write that as 13𝑖 plus 22𝑗.
We now have enough information to work out where the center of mass will be. We can follow a few steps to work out these coordinates. Firstly, we multiply every mass by its position vector. We then add up all these products and divide the total sum by the total mass. This is represented by the formula that the position vector 𝐑 of the center of mass of a system of masses is calculated by one over 𝑀, the total mass of the system, multiplied by the sum from 𝑖 equals one to 𝑛 of 𝑚 sub 𝑖 times vector 𝐫 sub 𝑖. 𝑚 sub 𝑖 is the mass of object 𝑖 and vector 𝐫 sub 𝑖 is the position vector of object 𝑖. So, let’s see how we can apply this to this system of masses.
We can work out the total mass to start. That’s six plus seven plus five plus nine plus eight grams, giving us a total of 35 grams. So, when we apply this formula, we have one over 35 outside the parentheses. We can then work out all the products in any order. So, let’s start with 𝐶 at position vector zero 𝑖 plus zero 𝑗. Its mass times its position vector will be nine times zero 𝑖 plus zero 𝑗. 𝐵 has a mass of five and a position vector of 26𝑖 plus zero 𝑗. For 𝐶, it’s a mass of six times a position vector of 26𝑖 plus 22𝑗. For the final two masses, we have eight times 39 plus 22𝑗 plus seven times zero 𝑖 plus 22𝑗.
We can then simplify by expanding across the parentheses, noting that the terms with zero 𝑖 or zero 𝑗 are simply zero. This leaves us with one over 35 times 130𝑖 plus 156𝑖 plus 132𝑗 plus 104𝑖 plus 176𝑗 plus 154𝑗. This can be simplified to one over 35 times 390𝑖 plus 462𝑗. Dividing 390𝑖 and 462𝑗 by 35 gives 78 over seven 𝑖 plus 66 over five 𝑗. But remember that we were asked for coordinates for the center of mass. So, we need to change this position vector for the center of mass into a set of coordinates. We can therefore give the answer that the coordinates of the center of mass are 78 over seven, 66 over five.
