Video Transcript
The figure shows a system of point
masses placed at the vertices of a square of side length six units. The mass placed at each point is
detailed in the table. Determine the coordinates of the
center of gravity of the system.
In the diagram, we have four point
masses, which we could also call particles, positioned at points 𝐴, 𝐵, 𝐶, and 𝐷,
which are the corners of a square. The table tells us that the point
mass at 𝐴 has a mass of 75 kilograms. The point mass at 𝐵 has a mass of
29 kilograms. And the masses at points 𝐶 and 𝐷
are 71 kilograms and 85 kilograms, respectively.
The question asks us to find the
center of gravity of the system. In this case, this is the same as
asking for the center of mass. We recall that the center of mass
is effectively the average position of all the mass in a system and that we can
calculate the 𝑥- and 𝑦-coordinates of the center of mass using these two
formulae.
Let’s begin by calculating the
𝑥-coordinate. We need to multiply each mass by
its corresponding 𝑥-coordinate, find the sum of these values, and then divide by
the sum of the masses. This can be written as shown. Since the square has side length
six units, if we let point 𝐴 lie at the origin, then the coordinates of points 𝐴,
𝐵, 𝐶, and 𝐷 are zero, zero; zero, six; six, six; and six, zero, respectively.
The mass at point 𝐴 is 75
kilograms. So we need to multiply this by
zero, the 𝑥-coordinate of point 𝐴. The mass at point 𝐵 is 29
kilograms. So we also need to multiply this by
zero, the 𝑥-coordinate of point 𝐵. For points 𝐶 and 𝐷, we have 71
multiplied by six and 85 multiplied by six, as both of these points have
𝑥-coordinates equal to six. The 𝑥-coordinate of the center of
mass is therefore equal to 75 multiplied by zero plus 29 multiplied by zero plus 71
multiplied by six plus 85 multiplied by six divided by 75 plus 29 plus 71 plus
85. This simplifies to 936 over 260,
which is equal to 18 over five. The 𝑥-coordinate of the center of
mass of the system is 18 over five or 3.6.
Let’s now consider the
𝑦-coordinate. This time, we need to multiply each
of the masses by their corresponding 𝑦-coordinate. We then find the sum of these
values and divide by the sum of the four masses. This is equal to 600 over 260,
which simplifies to 30 over 13. The 𝑦-coordinate of the center of
mass is 30 over 13. We can therefore conclude that the
center of gravity, or center of mass, of the system lies at the point 18 over five,
30 over 13.
