Finding the Center of Mass Using Double Integrals

[-EquinoX-]

1. Find the mass of a rod length 10 cm with density d(x) = e^-x gm/cm at a distance of x cm from the left.
2. Find the center of mass of a system containing three point masses of 5gm, 3gm, and 1 gm located respectively at x = -10, x = 1, and x =2.

for number two what I did is just this:

(5(-10) + 3(1) + 1(2)) / (-10+1+(-2)) and I got 45/7

 

[neutrino]

Hint For the first: Imagine the rod to be made of tiny mass elements dm. Each mass element is equal to \rho dx, where \rho is the linear density within the differential dx. If you sum up all such dm's you arrive at the mass of the rod.

2) How do you define, mathematically, the centre of mass of a system of particles?

 

[-EquinoX-]

so here's what I did for number one.

I take the integral of 0 to 10 of e^-x dx, is that all?

what's wrong with my number 2??

 

[neutrino]

so here's what I did for number one.

I take the integral of 0 to 10 of e^-x dx, is that all?

Yes, that's right

what's wrong with my number 2??

Again, I ask you, what's the definition of the centre of mass?

 

[-EquinoX-]

the center of mass as of my understanding is the point/position where there's a balance/equilibrium.

 

[neutrino]

the center of mass as of my understanding is the point/position where there's a balance/equilibrium.

I was talking about the mathematical definition, which is: \vec{R}_{cm} = \frac{\sum_{i=1}^{n}m_i\vec{r}_i}{\sum_{i=1}^n m_i}

If you look at your answer in the first post, you may notice that it is not dimensionally correct.

 

[-EquinoX-]

Oh clumsy mistake, so it should be:

(5(-10) + 3(1) + 1(2)) / (5+3+1)) and results as -45/9 = -5 right??

I have one more question and this is kind of hard:

A cardboard figure has a region which is bounded on the left by the line x = a, on the right by the line x=b, above by f(x), and below by g(x). If the density d(x) gm/cm^2 varies only with x, find an expression for the total mass of the figure, in terms of f(x), g(x), and d(x)

 

[neutrino]

I have one more question and this is kind of hard:

A cardboard figure has a region which is bounded on the left by the line x = a, on the right by the line x=b, above by f(x), and below by g(x). If the density d(x) gm/cm^2 varies only with x, find an expression for the total mass of the figure, in terms of f(x), g(x), and d(x)

The principle's the same as in post #2, except that, here you have an area instead of a line. Use double integrals.

 

[-EquinoX-]

The principle's the same as in post #2, except that, here you have an area instead of a line. Use double integrals.

what do you mean here as double integral?? so I still use the same formula as what I did in number 2?