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Finding centre of mass of a semicircular lamina

candy

New member
Mar 14, 2013
2
Suppose the density of any point on a semicircular lamina {(x, y) : x >= 0,\(\displaystyle x^2 + y^2\) <= r^2 is proportional to its distance from the origin. Compute the centre of mass.

I am trying to figure out what is the density function for this case.
 
Last edited:

MarkFL

Administrator
Staff member
Feb 24, 2012
13,775
Re: finding centre of mass of a semicircular lamina

The lamina here is the right half of a circular disk of radius $r$ centered at the origin.

What is the density function $\rho(x)$ and what kind of symmetry may we use?
 

candy

New member
Mar 14, 2013
2
Re: finding centre of mass of a semicircular lamina

The lamina here is the right half of a circular disk of radius $r$ centered at the origin.

What is the density function $\rho(x)$ and what kind of symmetry may we use?

Yaa.. thats what I am trying to figure out.. the density function.. and after that its pretty straight forward.
 

MarkFL

Administrator
Staff member
Feb 24, 2012
13,775
Re: finding centre of mass of a semicircular lamina

We are told the density at some point $P(x,y)$ is proportional to the point's distance from the origin. What is the distance of $P$ from the origin?
 

chisigma

Well-known member
Feb 13, 2012
1,704
Re: finding centre of mass of a semicircular lamina

Suppose the density of any point on a semicircular lamina {(x, y) : x >= 0,\(\displaystyle x^2 + y^2\) <= r^2 is proportional to its distance from the origin. Compute the centre of mass.

I am trying to figure out what is the density function for this case.
If the task is to find the coordinates x and y of the center of mass, the density function can be supposed to be...


$\displaystyle \delta (x,y)= \sqrt{x^{2}+ y^{2}} $ (1)

Remember that You have to perform the following steps...

a) compute the mass of the lamina...


$\displaystyle M= \int \int_{L} \delta (x,y)\ dx dy$ (2)


b) compute the coordinates of the center of mass...


$\displaystyle x_{g}= \frac{1}{M}\ \int \int_{L} x\ \delta(x,y)\ dx dy$

$\displaystyle y_{g}= \frac{1}{M}\ \int \int_{L} y\ \delta(x,y)\ dx dy$ (3)

Kind regards

$\chi$ $\sigma$
 
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HallsofIvy

Well-known member
MHB Math Helper
Jan 29, 2012
1,151
Re: finding centre of mass of a semicircular lamina

Suppose the density of any point on a semicircular lamina {(x, y) : x >= 0,\(\displaystyle x^2 + y^2\) <= r^2 is proportional to its distance from the origin. Compute the centre of mass.

I am trying to figure out what is the density function for this case.
Are you saying you do not know what "proportional" means? Or that you do not know what the distance from the origin is?

I would suggest changing to polar coordinates.