# Y coordinate of the system's center of mass?

• raptik
In summary, the problem involves finding the x and y coordinates of the center of mass for a system consisting of three uniform thin rods. The horizontal rod has a mass of 31 g and the two vertical rods each have a mass of 12 g. The x coordinate is found to be 14 cm, and using the same formula, the y coordinate is determined to be 16.8 cm.
raptik

## Homework Statement

In Fig. 9-39, three uniform thin rods, each of length L = 28 cm, form an inverted U. The vertical rods each have a mass of 12 g; the horizontal rod has a mass of 31 g. What are (a) the x coordinate and (b) the y coordinate of the system's center of mass? (Give your answer in cm)

http://edugen.wiley.com/edugen/courses/crs1650/art/qb/qu/c09/fig09_37.gif

## Homework Equations

xcom = (m1x1 + m2x2)/M

## The Attempt at a Solution

I got the x coordinate alright at 14cm but I can't find the y coordinate. I took the center of mass between the two vertical rods with y1 = 14cm and its m1 = 24g as total of the vertical rods. I then compared that center of mass to the center of mass of the horizontal rod with y2 = 28cm and its m2 = 31g. I thus found y = 21.89, but this is apparently wrong, could somebody please tell me what it is that I'm doing wrong?

Apply the same formula in the y direction.

2 vertical rods with com = L/2 and mass m = 2*(m*L/2) = mL

The horizontal rod at the top is m*L from the bottom, so the sum is 2m*L and with M = 3*m then

Com-y = 2*m*L/3*m = 2/3 L

What you said doesn't completely agree with the information, because the mass of the horizontal rod is different from the mass of the vertical rod. Also, where are you getting the mass of the rod with m = 2*(m*L/2)? Your explanation confuses me. I'm thinking that the mass is relative to the two masses so the m1 = 2(12g) and the y1 = L/2 which is 14cm. Is this wrong?

raptik said:
What you said doesn't completely agree with the information, because the mass of the horizontal rod is different from the mass of the vertical rod.
Sorry. Yes the horizontal rod has a different mass. I missed that. But the treatment is the same.
Also, where are you getting the mass of the rod with m = 2*(m*L/2)? Your explanation confuses me. I'm thinking that the mass is relative to the two masses so the m1 = 2(12g) and the y1 = L/2 which is 14cm. Is this wrong?

The vertical rods - 2 of them - each have the same mass and each have a Com at L/2 ... hence 2*(m(L/2))

So putting numbers to it then

m*L = .012*.28

And the horizontal rod is contributing a moment of .031*.28

Total then is (.012 + .031)*.28 /M = .033(.28)/.055 = 3/5(.28)

## 1. How is the Y coordinate of the system's center of mass calculated?

The Y coordinate of the system's center of mass is calculated by taking the sum of the products of each individual mass and its corresponding Y coordinate, then dividing by the total mass of the system.

## 2. What is the significance of the Y coordinate of the system's center of mass?

The Y coordinate of the system's center of mass is important because it represents the balance point of the system in the Y direction. This can be used to determine the overall stability and motion of the system.

## 3. How is the Y coordinate of the system's center of mass affected by changes in individual masses?

If the individual masses within the system are changed, the Y coordinate of the system's center of mass will also change. This is because the center of mass is a weighted average of the individual masses and their positions.

## 4. Can the Y coordinate of the system's center of mass ever be outside of the system?

No, the Y coordinate of the system's center of mass will always be within the boundaries of the system. This is because it is calculated using the positions of the individual masses within the system.

## 5. How does the Y coordinate of the system's center of mass relate to the X and Z coordinates?

The X, Y, and Z coordinates of the system's center of mass are all interrelated and can be used to fully describe the position of the center of mass within the system. However, each coordinate represents a different direction and should be considered separately when analyzing the motion and stability of the system.

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