Center of Mass coordinates

In summary, the composite slab has dimensions of d1=11.0cm, d2=2.80cm, and d3=13.0cm with half consisting of aluminum (density=2.70g/cm^3) and half consisting of iron (density=7.85g/cm^3). To find the center of mass, the volume for each piece was found to be 400.4g. The mass for aluminum was calculated to be 1081.08g and for iron it was 3143.17g. The center of mass position can be found along the direction of the 5.6cm thickness by using the mass and CM positions of the individual metals.
  • #1
ticklemepink20
1
0
1. A composite slab has dimensions of d1=11.0cm, d2=2.80cm, and d3=13.0cm. Half the slab consists of aluminum (density=2.70g/cm^3) and half consists of iron (density=7.85g/cm^3). What are the x coordinate, y coordinate, and z coordinate of the slab's center of mass?

There's a figure in my book where the dimensions are for each the iron and aluminum. In other words, if you find the volume using the dimensions they give you, if you double it you have the volume for the whole figure.


I have no idea where to even begin.
I found the volume for each piece to be 400.4g

I tried finding the mass for each using p=m/v.
For the aluminum I got m=(400.4cm^3)(2.70g/cm^3) = 1081.08g
For the iron I got m=(400.4cm^3)(7.85g/cm^3) = 3143.17g


No clue how to find x, y, and z...help please!
 
Physics news on Phys.org
  • #2
If you are saying that each metal is a rectangle 11cm x 13cm with a thickness of 2.8cm stacked to make a total thickness of 5.6cm, then by symmetry the CM position is on a line through the centers of the big ractangular faces. You just need to find the position along the direction of the 5.6cm thickness. You can do that by using the mass and the CM positions of the individual metals.
 
  • #3
[/b]

The center of mass coordinates for the composite slab can be calculated using the formula:

x = (m1x1 + m2x2) / (m1 + m2)
y = (m1y1 + m2y2) / (m1 + m2)
z = (m1z1 + m2z2) / (m1 + m2)

Where m1 and m2 are the masses of the aluminum and iron halves respectively, and x1, y1, z1 and x2, y2, z2 are the coordinates of the center of mass for each half.

To find the coordinates for each half, we can use the formula:

x = (d/2)cosθ
y = (d/2)sinθ
z = 0

Where d is the dimension of the half and θ is the angle between the dimension and the x-axis.

For the aluminum half, we have:
x1 = (11.0/2)cos(0) = 5.5cm
y1 = (11.0/2)sin(0) = 0cm
z1 = 0cm

For the iron half, we have:
x2 = (2.80/2)cos(0) = 1.4cm
y2 = (2.80/2)sin(0) = 0cm
z2 = 0cm

Substituting these values into the first set of equations, we get:
x = ((1081.08g)(5.5cm) + (3143.17g)(1.4cm)) / (1081.08g + 3143.17g) = 2.66cm
y = ((1081.08g)(0cm) + (3143.17g)(0cm)) / (1081.08g + 3143.17g) = 0cm
z = ((1081.08g)(0cm) + (3143.17g)(0cm)) / (1081.08g + 3143.17g) = 0cm

Therefore, the x coordinate of the slab's center of mass is 2.66cm, the y coordinate is 0cm, and the z coordinate is 0cm. This means that the center of mass lies on the x-axis,
 

What is the center of mass coordinate system?

The center of mass coordinate system is a mathematical concept used to describe the location and motion of an object or system of objects. It is based on the average position of all the mass in the system, and is often used in physics and engineering to simplify calculations and understand the behavior of objects.

How is the center of mass calculated?

The center of mass of an object or system can be calculated by finding the weighted average of the individual masses and their positions. This is typically done using the equation:
xcm = (m1x1 + m2x2 + ... + mnxn) / (m1 + m2 + ... + mn)
where xcm is the center of mass coordinate, mi is the mass at position xi, and n is the total number of masses in the system.

What is the significance of the center of mass?

The center of mass is significant because it is the point at which an object or system can be balanced. It also plays a crucial role in determining the overall motion and stability of an object or system. In addition, the center of mass can be used to simplify complex calculations by treating the entire object or system as a single point mass located at the center of mass.

How does the center of mass change with motion?

The center of mass will only change if there is an external force acting on the object or system. If there is no external force, the center of mass will remain in a constant position. However, if the object or system is in motion, the center of mass will also be in motion, following the path of the weighted average of all the individual masses.

Can the center of mass be located outside of an object?

Yes, the center of mass can be located outside of an object. This is often the case for irregularly shaped objects or systems with significantly uneven distributions of mass. In these cases, the center of mass may be located in empty space rather than within the physical boundaries of the object.

Similar threads

  • Introductory Physics Homework Help
Replies
8
Views
3K
  • Introductory Physics Homework Help
Replies
4
Views
908
  • Introductory Physics Homework Help
Replies
1
Views
549
  • Introductory Physics Homework Help
Replies
3
Views
4K
  • Introductory Physics Homework Help
Replies
9
Views
676
  • Introductory Physics Homework Help
Replies
19
Views
2K
  • Introductory Physics Homework Help
Replies
4
Views
449
  • Introductory Physics Homework Help
Replies
8
Views
2K
  • Introductory Physics Homework Help
Replies
10
Views
1K
  • Introductory Physics Homework Help
Replies
6
Views
3K
Back
Top