Three identical uniform meter sticks are placed on the floor. The first stick lies along the y axis from y = 0.320 m to y = 1.32 m. The second stick lies along the x axis from x = 0.330 m to x = 1.33 m. The third stick is positioned so that one end is on the x axis at x = 0.640 m, and the other end is on the y axis at y = 0.768 m. Calculate the location of the center of mass of the meter sticks.

since all the sticks are the same size and mass, the center is just the average of all the (x,y) midpoints:

(0,(.32+1.32)/2)
((.33+1.33)/2,0)
(.64/2,.768/2)

so, the center of mass is at

((0+(.33+1.33)/2+.64/2)/3,((.32+1.32)/2+0+.768/2)/3)
= (0.383,0.401)

Oh, the center of mass of meter sticks? Well, let's see if I can help you calculate that without tripping over my own calculations.

First, we need to find the masses of each meter stick. Since they are identical and uniform, we can assume that they have the same mass. Let's call it "m".

The first stick lies along the y-axis, from y = 0.320 m to y = 1.32 m, so its height is 1 meter. The second stick lies along the x-axis, from x = 0.330 m to x = 1.33 m, so its length is 1 meter as well. The third stick is positioned diagonally, so its height and length are both less than 1 meter.

To calculate the masses, we need to find the area of each stick. The area of a rectangular stick is its length multiplied by its height. So, the first two sticks have an area of 1 m² each.

The third stick is a bit tricky. We need to find the length and height of the triangular shape it forms. The length is the horizontal distance from (0.640 m, 0) to (0, 0.768 m), which is 0.640 m. The height is the vertical distance from (0, 0) to (0.640 m, 0.768 m), which is 0.768 m. So, the area of the third stick is (1/2) x 0.640 m x 0.768 m = 0.24576 m².

Now, let's calculate the total mass. Since all the sticks have the same mass, we can write:

m x (1 m² + 1 m² + 0.24576 m²) = 3m x 1.24576 m²

Now, to find the center of mass, we need to find the x-coordinate and y-coordinate separately.

The x-coordinate of the center of mass is given by the formula:

x_cm = (m₁x₁ + m₂x₂ + m₃x₃) / (m₁ + m₂ + m₃)

Where m₁, m₂, m₃ are the masses of the sticks, and x₁, x₂, x₃ are their respective x-coordinates.

Since the first stick lies along the y-axis, its x-coordinate is 0. For the second stick, it lies along the x-axis, so its x-coordinate is 0 as well. The third stick has one end on the x-axis at x = 0.640 m, so its x-coordinate is 0.640 m.

Therefore, the x-coordinate of the center of mass is:

x_cm = (m * 0 + m * 0 + m * 0.640 m) / (3m) = 0.640 m / 3 = 0.213 m

Now let's find the y-coordinate of the center of mass:

y_cm = (m₁y₁ + m₂y₂ + m₃y₃) / (m₁ + m₂ + m₃)

Where y₁, y₂, y₃ are the y-coordinates of the sticks.

For the first stick, the y-coordinate is (0.320 m + 1.32 m) / 2 = 0.820 m.
For the second stick, it lies along the y-axis, so its y-coordinate is 0.
For the third stick, one end is on the y-axis at y = 0.768 m, so its y-coordinate is 0.768 m / 2 = 0.384 m.

Therefore, the y-coordinate of the center of mass is:

y_cm = (m * 0.820 m + m * 0 + m * 0.384 m) / (3m) = (0.820 m + 0 + 0.384 m) / 3 = 0.454 m

So, the center of mass of the meter sticks is located at (0.213 m, 0.454 m).

To find the center of mass of the three meter sticks, we need to calculate the individual center of masses for each stick and then find the weighted average of these center of masses.

Let's start with the first stick along the y-axis. The length of this stick is (1.32 - 0.320) = 1 m, and its center of mass will be in the middle of its length, at 0.5 m from the bottom end.

Next, let's consider the second stick along the x-axis. The length of this stick is (1.33 - 0.330) = 1 m, and its center of mass will be in the middle of its length, at 0.5 m from the left end.

Now, for the third stick, we have one end on the x-axis and the other end on the y-axis. The x-coordinate of the center of mass will be at the midpoint between x = 0.640 m and x = 1 m, which is (0.640 + 1) / 2 = 0.82 m. Similarly, the y-coordinate of the center of mass will be at the midpoint between y = 0.768 m and y = 1.32 m, which is (0.768 + 1.32) / 2 = 1.044 m.

Finally, to find the overall center of mass, we need to calculate the weighted average of the center of mass coordinates, taking into account the lengths of the sticks.

Let the first stick have a mass of m1, the second stick have a mass of m2, and the third stick have a mass of m3.

Since all three meter sticks are identical, they have the same mass. So, m1 = m2 = m3.

The total length of the three sticks is 3 m, so the weight of each stick is 1/3 of the total mass.

Therefore, the center of mass coordinates are given by:

x_cm = (1/3) * (0.5 + 0.5 + 0.82)
y_cm = (1/3) * (0.5 + 0.5 + 1.044)

Calculating these values:
x_cm = (1/3) * (1.82) = 0.6067 m
y_cm = (1/3) * (2.044) = 0.6813 m

So, the location of the center of mass of the three meter sticks is approximately (0.6067 m, 0.6813 m).

To calculate the location of the center of mass of the meter sticks, we can use the concept of uniform distribution of mass along the sticks. We need to find the coordinates of the center of mass, which can be represented as (x_cm, y_cm).

First, let's find the mass of each meter stick. Since all three sticks are identical, they have the same mass.

To find the mass of each stick, we need to calculate its length. From the given information:

- The first stick lies along the y-axis and has a length of |y_final - y_initial| = |1.32 m - 0.320 m| = 1.0 m.
- The second stick lies along the x-axis and has a length of |x_final - x_initial| = |1.33 m - 0.330 m| = 1.0 m.

Both the first and second sticks have the same length of 1.0 m.

Now, let's calculate the mass of each stick using the formula mass = density * length, where density is assumed to be constant:

density = mass / length

Since the sticks are uniform, density is uniform too. Therefore, the mass of each stick is given by:

mass_of_each_stick = density * length

Next, let's calculate the coordinates of the center of mass.

The x-coordinate of the center of mass can be found using the formula:

x_cm = (m1*x1 + m2*x2 + m3*x3) / (m1 + m2 + m3),
where m1, m2, and m3 are the masses of the first, second, and third sticks, respectively, and x1, x2, and x3 are their respective x-coordinates.

Similarly, the y-coordinate of the center of mass can be found using the formula:

y_cm = (m1*y1 + m2*y2 + m3*y3) / (m1 + m2 + m3),
where y1, y2, and y3 are the respective y-coordinates.

Now, let's calculate the mass and coordinates of each stick:

mass_of_each_stick = density * length = density * 1.0 m = 1.0 * density.

Let's assign mass = mass_of_each_stick = 1.0 * density to all three sticks for simplicity.

Now, let's substitute the given values into the formulas:

x_cm = (m1*x1 + m2*x2 + m3*x3) / (m1 + m2 + m3)
= (mass*x1 + mass*x2 + mass*x3) / (3 * mass),
where x1, x2, and x3 are the respective x-coordinates.

Similarly,

y_cm = (m1*y1 + m2*y2 + m3*y3) / (m1 + m2 + m3)
= (mass*y1 + mass*y2 + mass*y3) / (3 * mass),
where y1, y2, and y3 are the respective y-coordinates.

Now, let's substitute the given values into the formulas:

x_cm = (mass*x1 + mass*x2 + mass*x3) / (3 * mass)
= (x1 + x2 + x3) / 3,

y_cm = (mass*y1 + mass*y2 + mass*y3) / (3 * mass)
= (y1 + y2 + y3) / 3.

Substituting the given coordinates into the formulas, we get:

x_cm = (0.330 m + 1.33 m + 0.640 m) / 3
= 2.3 m / 3,
= 0.767 m,

y_cm = (0.320 m + 1.32 m + 0.768 m) / 3
= 2.408 m / 3,
= 0.803 m.

Therefore, the center of mass is located at (x_cm, y_cm) = (0.767 m, 0.803 m).