three uniform thin rods, each of length L = 16 cm, form an inverted U. The vertical rods each have mass of 14 g; the horizontal rod has mass of 50 g. Where is the center of mass of the assembly (x, y)?
To find the center of mass of the assembly, we need to calculate the coordinates (x, y) where the center of mass is located.
Step 1: Find the center of mass of each individual rod.
Since all the rods are uniform and thin, the center of mass of each rod is at its midpoint.
For the vertical rods:
The two vertical rods have the same mass, so the center of mass for each vertical rod is at its midpoint, which is L/2 = 16 cm / 2 = 8 cm from the bottom end.
For the horizontal rod:
The horizontal rod has a mass of 50 g, and its length is also L = 16 cm. So, the center of mass of the horizontal rod is at its midpoint, which is L/2 = 16 cm / 2 = 8 cm from either end.
Step 2: Calculate the total mass and the x and y coordinates of the center of mass.
To find the x-coordinate of the center of mass, we need to add up the masses of each rod multiplied by their respective x-coordinate.
Let's denote the x-coordinate of the center of mass as x_cm.
Total mass = mass of vertical rod 1 + mass of vertical rod 2 + mass of horizontal rod
= 14 g + 14 g + 50 g = 78 g
For the vertical rods:
The x-coordinate of the center of mass for each vertical rod is 0 cm, as the rods are vertical and their x-coordinate does not contribute to the x-position of the center of mass.
For the horizontal rod:
The x-coordinate of the center of mass for the horizontal rod is L/2 = 8 cm.
Now we can calculate the x-coordinate of the center of mass, x_cm, using the formula:
x_cm = (mass of vertical rod 1 * x-coordinate of vertical rod 1 + mass of vertical rod 2 * x-coordinate of vertical rod 2 + mass of horizontal rod * x-coordinate of horizontal rod) / total mass
x_cm = (14 g * 0 cm + 14 g * 0 cm + 50 g * 8 cm) / 78 g
= (0 cm + 0 cm + 400 cm) / 78 g
= 400 cm / 78 g ≈ 5.13 cm
Therefore, the x-coordinate of the center of mass is approximately 5.13 cm.
The y-coordinate of the center of mass is the same as the y-coordinate of the midpoint of the horizontal rod, which is L/2 = 8 cm from the bottom end.
So, the coordinates of the center of mass of the assembly are (x, y) ≈ (5.13 cm, 8 cm).