Center of Gravity? (physics problem)?

Question

Find the x- and y-coordinates of the center of gravity of a 16-m by 16-m uniform sheet of plywood with a rectangle, that is 11-m by 5-m, removed from the upper left hand corner of the plywood. How do you find this??

how can you solve this problem mathematically with physics??

please help me understand hooooow to solve this problem. thanks~

Answer 1

Let ABCD be the square with
AB =16m at the top and
CD =16m at the bottom.
AC= 16m at the left.
BD = 16m at the right.

Also let
AE = 11m along AB and
AF = 5m along AC.

The center of gravity of the full square is it at its center, i.e., at 8, 8. The area of the square is 16 x 16 = 256

The center of gravity of the rectangle is 5.5, 13.5 from the bottom. Its area = 11x5 = 55

Taking moment of the areas about the line AC

[256 x 8 - 55 x 5.5] / [256 - 55] = 8.68 m from the side AC.

Taking moment of the areas about the line CD.

[256 x 8 - 55 x 13.5] / [256 - 55] = 6.46 m from CD

Therefore the C. g is at 8.68 m from the side AC and 6.46 m from CD

Answer 2

Use superposition.

You know that the center of mass of a uniform sheet is in the middle, right? For a 16x16 sheet, that point is at (8,8) if the lower left corner of the sheet is at the origin.

You don't say whether the 11x5 piece is a vertical or horizontal chunk, I'll take it to be horizontal, that is 11m along x, and 5m along y. The center of this piece is at (11/2, 5/2 + 11) or (11/2, 27/2). Now add the two centers of mass according to the center of mass formula . . . but SUBTRACT for the piece that ISN'T there (the last bit is to find the mass of that piece relative to the whole):

Rcm = (16,16) - (11/2, 27/2)*(11*5)/(16*16)

Rcm = (14.8, 13.1)m

Answer 3

Consider the sheet to have mass m
thus the mass per area is m/256
Now if a piece 11 by 5 m is taken away, then the mass removed is 55m/256 hence mass remaining is m-[55m/256]
Now consider the part remaining
It is made up of one square and a rectangle
Square is 11 by 11 m hence area is 121 sq.m
Rectangle is of 16 by 5 m hence area is 80 sq.m
hence their masses are 121m/256 and 80m/256

Now find their COMs and then assuming them to be point particles, use the formula
X co-ordinate of COM of rem plate= [m1x1+m2x2] / {m-[55m/256]}
Similarly Y=[m1y1+m2y2] / {m-[55m/256]}

m will cancel out and you will get a numerical value

Hope that answers your question

Answer 4

Put a pin in the plywood, have the pin attatched to something like a wall.

The plywood should be able to easily swing from side to side with the pin being the pivot, leave the plywood to come to rest. then make a vertical line from where the pin is.

Do this again in another place. Where the lines cross gives you the centre of gravity.

Answer 5

It is 2.5 from top and bottom and 5.5 from sides.
Or X= 2.5 , Y= 5.5 and X= - 2.5 , Y= - 5.5 in coordinate system of 16mX16m from 0 point which is 8m from each side.

Answer 6

Find the center, since the mass is uniformly distributed this is the same as the centre of gravity.

Answer 7

You can do an experiment.