Density and integrals

In summary: What do you mean here as double integral?? so I still use the same formula as what I did in number 2?Yes, that's right, you use a double integral.
  • #1
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1
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
 
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  • #2
Hint For the first: Imagine the rod to be made of tiny mass elements [itex]dm[/itex]. Each mass element is equal to [itex]\rho dx[/itex], where [itex]\rho[/itex] is the linear density within the differential [itex]dx[/itex]. If you sum up all such [itex]dm[/itex]'s you arrive at the mass of the rod.

2) How do you define, mathematically, the centre of mass of a system of particles?
 
  • #3
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??
 
  • #4
-EquinoX- said:
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?
 
  • #5
the center of mass as of my understanding is the point/position where there's a balance/equilibrium.
 
  • #6
-EquinoX- said:
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: [tex]\vec{R}_{cm} = \frac{\sum_{i=1}^{n}m_i\vec{r}_i}{\sum_{i=1}^n m_i}[/tex]

If you look at your answer in the first post, you may notice that it is not dimensionally correct.
 
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  • #7
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)
 
  • #8
-EquinoX- said:
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.
 
  • #9
neutrino said:
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?
 

1. What is density?

Density is a measure of how much mass is contained in a given volume. It is often represented by the Greek letter rho (ρ) and is typically measured in units of mass per unit volume, such as grams per cubic centimeter (g/cm3) or kilograms per cubic meter (kg/m3).

2. How is density calculated?

Density is calculated by dividing the mass of an object by its volume. The formula for density is: ρ = m/V, where ρ is density, m is mass, and V is volume.

3. What is the relationship between density and buoyancy?

Density plays a crucial role in determining whether an object will sink or float in a fluid. If the density of an object is greater than the density of the fluid it is placed in, it will sink. If the density of an object is less than the density of the fluid, it will float.

4. What is an integral?

An integral is a mathematical concept that represents the area under a curve. It is used to calculate the total value of a function over a given interval. Integrals are often used in physics and engineering to solve problems related to motion, force, and energy.

5. How are density and integrals related?

Density and integrals are related in that the density of an object can be calculated by taking the integral of its mass distribution over its volume. In other words, the total mass of an object can be found by integrating its density function over its volume. This relationship is often used in physics and engineering to determine the mass of irregularly shaped objects.

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