Specify the position of the center of mass

[nayfie]

Homework Statement

Two identical uniform rectangular flat planks with sides a and b are glued together to form a T-shaped object. Specify the position of the center of mass. State any theorems you use in order to arrive at your conclusion. See the attempted solution for a diagram. :)

The problem:

I have shown that the center of mass for each object is ([itex]\frac{a}{2}, \frac{b}{2}[/itex]), but I can't seem to show (using maths) that the center of mass of the overall object is [itex](\frac{\frac{a}{2} + \frac{b}{2}}{2})[/itex].

If anybody could help out that would be wonderful. :)

Homework Equations

[itex]CM = \int^{x}_{0}(x\frac{dm}{dx})dx[/itex]

The Attempt at a Solution

It would probably take me a good 30 minutes to write all of this out, so I'll just attach a picture. :)

http://dl.dropbox.com/u/29493853/Photo%20Jul%2031%2C%203%2020%2014%20PM.jpeg"

 

[PeterO]

Homework Statement

Two identical uniform rectangular flat planks with sides a and b are glued together to form a T-shaped object. Specify the position of the center of mass. State any theorems you use in order to arrive at your conclusion. See the attempted solution for a diagram. :)

The problem:

I have shown that the center of mass for each object is ([itex]\frac{a}{2}, \frac{b}{2}[/itex]), but I can't seem to show (using maths) that the center of mass of the overall object is [itex](\frac{\frac{a}{2} + \frac{b}{2}}{2})[/itex].

If anybody could help out that would be wonderful. :)

Homework Equations

[itex]CM = \int^{x}_{0}(x\frac{dm}{dx})dx[/itex]

The Attempt at a Solution

It would probably take me a good 30 minutes to write all of this out, so I'll just attach a picture. :)

http://dl.dropbox.com/u/29493853/Photo%20Jul%2031%2C%203%2020%2014%20PM.jpeg"

Would you find it simpler if you had two spheres of lead, positioned so that their centres of mass were in equivalent positions? ie the same distance apart?

 

[I like Serena]

Hi nayfie!

A center of mass is a point with an x-coordinate and a y-coordinate.
To state where it is, first you need to define an origin.
What is your origin?

The answer you are trying to find is just a value, while it should be a point.
And either way, I'm afraid that the value will not be part of the answer.
Or were the planks supposed to be glued on top of each other, instead of against each other?

The relevant equation you specified is not the one usually used for a CM calculation.
Where did you get it?
There are several assumptions contained within that may block you.
For one it assumes the object is between a coordinate 0 and x, meaning it is entirely to the right of the origin.
Furthermore it only calculates the x coordinate of the CM.
What happened to the y coordinate?

Btw, are you required to calculate it using integrals?
For there is an easier method, since a CM is just a mass-weighted average of all objects...