College Physics- Center of Mass?
The objects in the figure below are constructed of uniform wire bent into the shapes shown (Figure 1) . Each peice has a length of 0.15 .
There are two objects that we need to figure out the x- and y-coordinates of the center of mass of the object . Assume the origin to be at the bottom left point.
Help please
1. the first object is like a l_l the wires are connected in this exact shape.
2. is the shape of L
3. is an equilateral triangle.
Remember each side is 0.15m .. help me please
1. On axis of symmetry midway between vertical wires and 0.05 m above bottom wire.
2. Along a 45 degree from the corner of the L. 0.0375 m to the right of and 0.0375 m above the corner.
3. At the center of the triangle. (The center of symnmetry where the angle bisectors intersect.)
Yes but how do we go about this.
Let me show you what I did for l_l
r(C) = Σ{r(i) •m(i)}/Σm(i)
Assume the origin to be at the bottom left point and the mass of each part is ’m’
1. X(C) ={ 0+(L/2)+L}m/3m = L/2
Y(C) = { (L/2) +0+(L/2)}m/3m = L/3
C= (L/2; L/3) = (0.075;0.05).
2. X(C) ={ 0+(L/2)}m/2m = L/4
Y(C) = { (L/2) +0}m/2m = L/4
C= (L/4; L/4) = (0.0375;0.0375).
3. X(C) ={ L/2•cos60°)+(L/2)+(L- L/2•cos60°)}m/3m = L/2
Y(C) = { (L/2•sin60°) +0+(L/2•sin60°))}m/3m = 0.29L
C= (L/2; L/3) = (0.075;0.0435).
To find the coordinates of the center of mass of these objects, you need to calculate the weighted average of the x and y coordinates of all the individual points that make up the object. Here's how you can find the coordinates for each of the given objects:
1. Object like L:
- Divide the object into smaller segments and assign a mass to each segment proportional to its length.
- For this object, the segments can be divided into three parts: the vertical segment, the horizontal segment, and the diagonal segment.
- The vertical and horizontal segments have lengths of 0.15 m each, and the diagonal segment has a length of √(0.15^2 + 0.15^2) = 0.2121 m.
- Assign a mass of 0.15 kg to each 0.15 m segment, and a mass of 0.2121 kg to the diagonal segment.
- The x-coordinate of the center of mass can be calculated using the formula: (m1 * x1 + m2 * x2 + m3 * x3) / (m1 + m2 + m3), where m1, m2, and m3 are the masses of the segments, and x1, x2, and x3 are their respective x-coordinates.
- Let's assume the x-coordinate of the leftmost point is 0. Then, the x-coordinate of the center of mass is: (0.15 * 0 + 0.2121 * 0.15 + 0.15 * 0.15) / (0.15 + 0.2121 + 0.15).
- Calculate this expression to find the x-coordinate.
- Similarly, you can calculate the y-coordinate using the formula: (m1 * y1 + m2 * y2 + m3 * y3) / (m1 + m2 + m3), where y1, y2, and y3 are the y-coordinates of the respective segments.
2. L-shaped object:
- Divide the object into two segments: the horizontal segment and the vertical segment.
- Both segments have lengths of 0.15 m.
- Assign a mass of 0.15 kg to each segment.
- The x-coordinate of the center of mass can be calculated using the formula mentioned earlier.
- Assuming the x-coordinate of the leftmost point is 0, calculate the x-coordinate of the center of mass.
- Calculate the y-coordinate in a similar manner.
3. Equilateral triangle:
- Divide the triangle into three equal segments.
- Each segment has a length of 0.15 m.
- Assign a mass of 0.15 kg to each segment.
- Since the triangle is symmetrical, the x-coordinate of the center of mass is 0.
- Calculate the y-coordinate using the formula.
Remember to convert the units between meters and kilograms to be consistent in your calculations.