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 determine the center of mass of the assembly, we need to find the position (x, y) where the resultant gravitational forces acting on the system balance out.

Let's break down the problem and consider the center of mass for each individual rod:

1. Vertical Rods:
- Each vertical rod has a length of L = 16 cm and a mass of 14 g.
- The center of mass for each vertical rod is at its midpoint, which is L/2 = 8 cm away from the bottom end of the rod.
- Since there are two vertical rods, the combined center of mass for both vertical rods is at the midpoint between their centers of mass, which is L/2 = 8 cm away from the bottom end of the rods.

2. Horizontal Rod:
- The horizontal rod has a length of L = 16 cm and a mass of 50 g.
- The center of mass for the horizontal rod is at its midpoint, which is L/2 = 8 cm away from either end of the rod.

Now, let's find the overall center of mass for the entire assembly:

Since the two vertical rods contribute an equal amount of mass on each side, their combined center of mass will lie in the middle, horizontally. Therefore, the x-coordinate of the center of mass will be the same as the x-coordinate of the midpoint between the two vertical rods' centers of mass.

The y-coordinate of the center of mass will be the weighted average of the individual center of masses of the vertical rods and the horizontal rod. We will consider the mass of each rod as the weight for the average.

Using this information, the x, y coordinates of the center of mass can be calculated as follows:

x-coordinate of center of mass = x-coordinate of midpoint between vertical rod centers of mass = 8 cm

y-coordinate of center of mass = ([(mass of vertical rods) * (y-coordinate of vertical rods' center of mass)] + (mass of horizontal rod) * (y-coordinate of horizontal rod's center of mass)) / (total mass of assembly)

y-coordinate of center of mass = [(2 * 14 g) * 8 cm + 50 g * 8 cm] / (2 * 14 g + 50 g)

Simplifying the equation:

y-coordinate of center of mass = (308 g*cm + 400 g*cm) / 78 g

y-coordinate of center of mass = 7.18 cm

Therefore, the center of mass of the assembly is located at the coordinates (8 cm, 7.18 cm) or (x, y).

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