# The z coordinate of the center of mass of the box

• hateAleen
In summary, the problem involves finding the center of mass of a cubical box made of uniform metal plate with a length of 97 cm. The formula for finding the center of mass is rcom=(1/M)sum(mi*ri), where M represents the total mass and ri represents the position vector of each mass. To find M, the mass of each side can be assumed to be equal to M, and the center of mass for the four upright sides can be treated as a single particle with a mass of 4M. Combining this with a particle of mass M in the middle of the bottom allows for the calculation of the overall center of mass.
hateAleen

## Homework Statement

A cubical box has been constructed from uniform metal plate of negligible thickness. The box is open at the top and has edge length L = 97 cm. Find (a) the x coordinate, (b) the y coordinate, and (c) the z coordinate of the center of mass of the box.

## Homework Equations

rcom=(1/M)sum(mi*ri)

## The Attempt at a Solution

I could not figure out what to do with a three dimensional object since we are not given a mass.

hateAleen said:

## The Attempt at a Solution

I could not figure out what to do with a three dimensional object since we are not given a mass.

Assume some $\sigma$ as the two dimensional mass density, that is, mass per unit area of the faces. Multiplying $\sigma$ by area, you'll get the mass. I think you know where the CM of a rectangle is. Now, apply the formula you know.

If you assume that the mass of one side is M then the com of the four upright sides will correspond to a "particle" of mass 4M in the middle of the box. You now need to combine this with a "particle" of mass M in the middle of the bottom to get the overall com.

## 1. What is the z coordinate of the center of mass of the box?

The z coordinate of the center of mass of the box is a numerical value that represents the vertical position of the center of mass. It is typically measured in meters or feet, depending on the units used for the dimensions of the box.

## 2. How is the z coordinate of the center of mass calculated?

The z coordinate of the center of mass is calculated by finding the average of the z coordinates of all the individual points that make up the box. This can be done by multiplying each z coordinate by its corresponding mass and then dividing the sum of these products by the total mass of the box.

## 3. Why is the z coordinate of the center of mass important?

The z coordinate of the center of mass is important because it helps determine the stability and equilibrium of the box. Objects with a lower z coordinate of the center of mass are more stable and less likely to tip over compared to those with a higher z coordinate.

## 4. Can the z coordinate of the center of mass change?

Yes, the z coordinate of the center of mass can change if the dimensions or mass distribution of the box is altered. For example, if a heavy object is added to one side of the box, the center of mass will shift towards that side, resulting in a change in the z coordinate.

## 5. How does the z coordinate of the center of mass affect the box's motion?

The z coordinate of the center of mass affects the box's motion by influencing its stability and tipping point. A lower z coordinate results in a more stable box and a higher z coordinate increases the likelihood of the box tipping over. This can also impact the box's rotational and translational motion.

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