How to find center of gravity?

In summary, the problem involves finding the x- and y-coordinates of the center of gravity of a uniform sheet of plywood with a cutout in the upper right quadrant. The dimensions of the cutout are given as a = 4.30 ft and b = 1.70 ft. The solution involves splitting the plywood into two sections, A and B, and finding the centers of gravity for each section. The x-coordinate for the center of gravity for section B was miscalculated due to using the incorrect dimension of 3.7 ft instead of 4.3 ft. The correct calculation for the x-coordinate is 3.612 ft. The final step is to find the center of gravity of the entire plywood sheet by
  • #1

Homework Statement



Find the x- and y-coordinates of the center of gravity of a 4.00 ft by 8.00 ft uniform sheet of plywood with the upper right quadrant removed as shown in the figure below. The dimensions of the cutout are a = 4.30 ft and b = 1.70 ft.

I found:

xA= 3.7/2 cm
xB=3.7+(4.3/2) cm
Ya=2 cm
Yb=2.3/2 cm
Ycm (Y-coordinates)= 1.689 cm

Homework Equations



xcm=maxa+mbxb/ma+mb
ycm=maya+mbyb/ma+mb
ma=(mu)A
mb=(mu)B
m=ma+mb
mass of a= area of A/total area
mass of b= area of B/total area

The Attempt at a Solution



I know the basic steps:
I first 'split' the plywood into two:
A (area: 4 * 3.7 cm^2)
and
B (area: 3.7 * 2.3 cm^2)

I find the centers of gravity of both A and B, and want to find what is the intermediate center of gravity of the whole piece. As I've said before, I've found the y-coordinate of the center of gravity, but somehow miscalculated the x-coordinate.

I've got the y-coordinates (which is 1.689), but somehow I think I made an error in the calculation of the x-coordinates which I calculated as somewhere around 3.3117-3.3118 or 3.312.
 
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  • #2
Please get rid of that 'cm' unit..the problem is in feet... that's the way I order my plywood, in 4' X 8' sheets :smile:. I haven't checked your numbers, but I note that in calculating ther area of B, you used 3.7 (ft) as one of the dimensions when it should be 4.3 ft... and that of course is feet not centimeters ...:wink:
 
  • #3
That didn't help either It calculated as 3.612, but that was wrong as well.
 
  • #4
I guess the usual language in which you solve this is - Consider the full plywood 'minus' the cutout. Choose an origin, find center of gravity of the whole plywood, center of gravity of cutout with a negative mass and find the center of gravity of the two using the equations you wrote in OP.

If you are not comfortable with this language, a more detailed/clear answer can be formed, I'm sleepy >.>
 

1. What is the center of gravity?

The center of gravity is the point where the entire weight of an object is concentrated. It is the point at which the object would balance if it were suspended at that particular point.

2. How do you calculate the center of gravity?

The center of gravity can be calculated by finding the weighted average of all the individual points of an object. This is done by multiplying the weight of each point by its distance from a chosen reference point and then dividing the sum of these values by the total weight of the object.

3. Why is it important to find the center of gravity?

Finding the center of gravity is important for understanding the stability and balance of an object. It is also crucial in engineering and design as it helps determine the placement of components to maintain stability and prevent tipping or overturning.

4. Can the center of gravity change?

Yes, the center of gravity can change if the distribution of weight within an object changes. For example, if you add or remove weight from one side of an object, the center of gravity will shift accordingly.

5. How can you find the center of gravity for irregularly shaped objects?

For irregularly shaped objects, the center of gravity can be found by using a plumb line or by balancing the object on a pivot point. Alternatively, it can also be calculated by using mathematical formulas specific to the shape of the object.

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