find center of mass of 2m thin rod, thin plate w/ y=16-x^2 &

[atul]

Can anyone help me with these problems for center of mass . thanks for helping
1. find the center of mass of a thin rod of length 2m with density

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[/img]= 2x
if the image doesnt show up its 8(not exactly 8 its the symbol thats missing right half circle on top) = 2x

2. find the center of mass of a thin plate of uniform density which is defined by y=16-x^2 and the x-axis.

3. find the center of mass of a thin plate of uniform density which is defined by y=9-x^2 and the line y=x+3

 

[royhaas]

Re: center of mass problems help..

A center of mass for something with a uniform density (in two dimensions) is given by \(\displaystyle \frac{\int xf(x)dx}{\int f(x)dx}\), where the function \(\displaystyle f(x)\) describes the shape of the object along with the limits of integration to define the boundaries. Drawing a picture of the region \(\displaystyle f(x)\)is usually a good way to start.

 

[atul]

Re: center of mass problems help..

tried but these 3 problems are killing me.

 

[atul]

Can anyone help me with these problems for center of mass . thanks for helping
1. find the center of mass of a thin rod of length 2m with density 8(not exactly 8 its the symbol thats missing right half circle on top) = 2x.

2. find the center of mass of a thin plate of uniform density which is defined by y=16-x^2 and the x-axis.

3. find the center of mass of a thin plate of uniform density which is defined by y=9-x^2 and the line y=x+3

I posted these questions in calculus noone was able to help someone tried to help me get started but i still am having difficulty with these three problems. can anyone help me out? help would be appriciated much. thanks.

 

[Deleted member 4993]

Can anyone help me with these problems for center of mass . thanks for helping
1. find the center of mass of a thin rod of length 2m with density 8(not exactly 8 its the symbol thats missing right half circle on top) = 2x.

2. find the center of mass of a thin plate of uniform density which is defined by y=16-x^2 and the x-axis.

3. find the center of mass of a thin plate of uniform density which is defined by y=9-x^2 and the line y=x+3

I posted these questions in calculus noone was able to help someone tried to help me get started but i still am having difficulty with these three problems. can anyone help me out? help would be appriciated much. thanks.

The reason nobody answered your post is - you have not followed instructions spelled out in "read before posting".

You need to show your work - beyond just posting the problem and declaring "I need help".

Please show your work for each of the problems, indicating exactly where you are stuck - so that we know where to begin to help you.

 

[stapel]

Duplicate threads merged.

 

[galactus]

2. find the center of mass of a thin plate of uniform density which is defined by \(\displaystyle y=16-x^{2}\) and the x-axis.

The center of mass is the point where this would balance if we put it on, say, the end of a pencil.

We have a concave down parabola. Because the center of mass lies on its axis of symmetry, \(\displaystyle \overline{x}=0\)

All remains is finding \(\displaystyle \overline{y}\).

The mass is \(\displaystyle m={\rho}\int_{-4}^{4}(16-x^{2})dx={\rho}\frac{256}{3}\)

We can also get this by just using the formula for the area of a parabola. \(\displaystyle A=\frac{2}{3}bh=\frac{2}{3}\cdot 16\cdot 8=\frac{256}{3}\)

To find the moment about the x axis, place a 'representative' rectangle in the region. The distance from the x axis to the center of the rectangle is

\(\displaystyle y_{k}=\frac{f(x)}{2}=\frac{16-x^{2}}{2}\)

Because the mass of the rectangle is \(\displaystyle {\rho}f(x){\Delta}x={\rho}(16-x^{2}){\Delta}x\)

We have:

\(\displaystyle M_{x}={\rho}\int_{-4}^{4}\frac{16-x^{2}}{2}(16-x^{2})dx\)

\(\displaystyle \overline{y}\) is given by \(\displaystyle \frac{M_{x}}{m}\)

Finish?. The center of mass is at \(\displaystyle (0, \;\ \text{result from above})\)