How can I calculate the cumulative mass of a disk using disk mass density?

[independentphysics]

Homework Statement

Find cumulative mass of a disk

Relevant Equations

Exponential mass function

I want to find the cumulative mass m(r) of a mass disk. I have the mass density in terms of r, it is an exponential function:
ρ(r)=ρ0*e^(-r/h)

A double integral in polar coordinates should do, but im not sure about the solution I get.

 

[Mark44]

Homework Statement: Find cumulative mass of a disk
Relevant Equations: Exponential mass function

I want to find the cumulative mass m(r) of a mass disk. I have the mass density in terms of r, it is an exponential function:
ρ(r)=ρ0*e^(-r/h)

A double integral in polar coordinates should do, but im not sure about the solution I get.

What solution do you get?

Also, what is the significance of h in your equation?

You can write an iterated integral in LaTeX like this: $\int_{\theta = 0}^b \int_{r = 0}^a f(r, \theta)~dr~d\theta$

 

[independentphysics]

What solution do you get?

Also, what is the significance of h in your equation?

You can write an iterated integral in LaTeX like this: $\int_{\theta = 0}^b \int_{r = 0}^a f(r, \theta)~dr~d\theta$

I am getting: ρ(r)=2*π*ρ0 * h^2 * [1 - e^(-r/h)]
h is a constant

 

[pasmith]

Is your intermediate result $2\pi \rho_0 \int_0^R re^{-r/h}\,dr$?

 

[independentphysics]

Is your intermediate result $2\pi \rho_0 \int_0^R re^{-r/h}\,dr$?

Yes, it is.

 

[Mark44]

Is your intermediate result $2\pi \rho_0 \int_0^R re^{-r/h}\,dr$?

Yes, it is.

Assuming @pasmith's work is correct (I haven't worked the problem), the integral shown can be evaluated using integration by parts.

 

[independentphysics]

Assuming @pasmith's work is correct (I haven't worked the problem), the integral shown can be evaluated using integration by parts.

Yes, I am aware of it, but Im not sure about my final result.

 

[Mark44]

Yes, I am aware of it, but Im not sure about my final result.

Well, why don't you show us what you got, together with your work?

 

[independentphysics]

Well, why don't you show us what you got, together with your work?

From $2\pi \rho_0 \int_0^R re^{-r/h}\,dr$

I evaluate this integral using integration by substitution. Let u = r/h, then du/dr = 1/h and dr = h*du. Substituting, we get:

m(r) = 2π * ρ0 * h^2 * ∫ u * e^(-u) * du

Is this correct?

 

[Mark44]

From $2\pi \rho_0 \int_0^R re^{-r/h}\,dr$

I evaluate this integral using integration by substitution. Let u = r/h, then du/dr = 1/h and dr = h*du. Substituting, we get:

m(r) = 2π * ρ0 * h^2 * ∫ u * e^(-u) * du

Is this correct?

It's technically correct, but it doesn't get you any closer to evaluating the integral. To integrate $\int re^{-r/h}dr$ you need to use integration by parts, something I mentioned before.
A substitution won't be useful here.

 

[independentphysics]

It's technically correct, but it doesn't get you any closer to evaluating the integral. To integrate $\int re^{-r/h}dr$ you need to use integration by parts, something I mentioned before.
A substitution won't be useful here.

How do I integrate by parts the function?

 

[Mark44]

How do I integrate by parts the function?

Your calculus textbook should have a section on integration by parts, along with several examples.

 

[independentphysics]

Your calculus textbook should have a section on integration by parts, along with several examples.

If I choose u=r and v=e^(-r/h) so that du=1, will this do?

 

[SammyS]

If I choose u=r and v=e^(-r/h) so that du=1, will this do?

No.

If $u=r$ (that's fine), then $du=dr$.

For $v$, you need $\displaystyle dv=e^{-r/h}\,dr$.

Integrate to get $v$.

 

[independentphysics]

No.

If $u=r$ (that's fine), then $du=dr$.

For $v$, you need $\displaystyle dv=e^{-r/h}\,dr$.

Integrate to get $v$.

ρ(r)= ρ0*e^(-r/h)

2π results from integration of angle in polar coordinates

integrate (r*e^(-r/h) dr)
u=r
du=dr
dv=e^(-r/h)*dr
v=-h*exp(-r/h)

ρ(r)=2π * ρ0 * ( r*h*(-e^(-r/h) + int(h*e^(-r/h)dr )

Is it correct?

 

[Mark44]

ρ(r)= ρ0*e^(-r/h)

2π results from integration of angle in polar coordinates

integrate (r*e^(-r/h) dr)
u=r
du=dr
dv=e^(-r/h)*dr
v=-h*exp(-r/h)

ρ(r)=2π * ρ0 * ( r*h*(-e^(-r/h) + int(h*e^(-r/h)dr )

Is it correct?

It looks like you're on the right track. The biggest mistake I see is that your equation starts off with $\rho(r)$ when it should be m(r), the cumulative mass. Again, I haven't worked through the problem, and am assuming that @pasmith's hint is correct.

If so, all you need to do to finish is 1) evaluate $\int h e^{-r/h}dr$, and then 2) evaluate the resulting large expression at r = 0 and r = R, subtracting the first expression (with r = 0) from the latter expression.

The integral I mentioned is relatively straightforward -- a simple substitution will work.

 

[independentphysics]

It looks like you're on the right track. The biggest mistake I see is that your equation starts off with $\rho(r)$ when it should be m(r), the cumulative mass. Again, I haven't worked through the problem, and am assuming that @pasmith's hint is correct.

If so, all you need to do to finish is 1) evaluate $\int h e^{-r/h}dr$, and then 2) evaluate the resulting large expression at r = 0 and r = R, subtracting the first expression (with r = 0) from the latter expression.

The integral I mentioned is relatively straightforward -- a simple substitution will work.

Equation:
ρ(r)=ρ0*e^(-r/h)
In polar coordinates, a 2π factor multiplies the integratal (r*e^(-r/h) dr)
u=r
du=dr
dv=e^(-r/h)*dr
v=-h*exp(-r/h)
2π * ρ0 * ( r*h*(-e^(-r/h) + int(h*e^(-r/h)dr )
M (r)=2π * ρ0 * h * (h + r) * (-e^(-r/h)) + c
I find c by M(0)=c

 

[Mark44]

Equation:
ρ(r)=ρ0*e^(-r/h)
In polar coordinates, a 2π factor multiplies the integratal (r*e^(-r/h) dr)
u=r
du=dr
dv=e^(-r/h)*dr
v=-h*exp(-r/h)
2π * ρ0 * ( r*h*(-e^(-r/h) + int(h*e^(-r/h)dr )
M (r)=2π * ρ0 * h * (h + r) * (-e^(-r/h)) + c
I find c by M(0)=c

Your final answer for M should involve h and R, but should not involve r.
You need to evaluate M(R) and M(0), and your final answer for M will be M(R) - M(0).
For M(0) I get $2\pi \rho_0 h^2$.

 

[independentphysics]

Your final answer for M should involve h and R, but should not involve r.
You need to evaluate M(R) and M(0), and your final answer for M will be M(R) - M(0).
For M(0) I get $2\pi \rho_0 h^2$.

The final answer for cumulative mass M(r) and depends on r.

R is end of the disk, which doesn't even need to exist (I've never mentioned if the disk in infinite).

M(0) is M(r=0).

The final answer for cumulative mass M(r) involves +constant an the end of the discussed integral, which is M(0).

 

[pasmith]

The mass of the disc $0 \leq r < R$ is $$M(R) = 2\pi\rho_0 \int_0^R re^{-r/h}\,dr.$$ It follows immediately that $M(0) = 0$. Since $$\frac{d}{dr} re^{-r/h} = (1 - rh^{-1})e^{-r/h}$$ we have $$re^{-r/h} = he^{-r/h} - h\frac{d}{dr} re^{-r/h}$$ and thus $$\int_0^R re^{-r/h}\,dr = h^2(1 - e^{-R/h}) - hRe^{-R/h}.$$

 

[independentphysics]

The mass of the disc $0 \leq r < R$ is $$M(R) = 2\pi\rho_0 \int_0^R re^{-r/h}\,dr.$$ It follows immediately that $M(0) = 0$. Since $$\frac{d}{dr} re^{-r/h} = (1 - rh^{-1})e^{-r/h}$$ we have $$re^{-r/h} = he^{-r/h} - h\frac{d}{dr} re^{-r/h}$$ and thus $$\int_0^R re^{-r/h}\,dr = h^2(1 - e^{-R/h}) - hRe^{-R/h}.$$

There is a constant (+C) after integrating the density function to obtain cumulative mass M(r), and M(0) is not 0.

 

[pasmith]

There is a constant (+C) after integrating the density function to obtain cumulative mass M(r), and M(0) is not 0.

You seem confused. There is no arbitrary constant in a definite integral, which is what $M(R) = 2\pi\int_0^R r\rho(r)\,dr$ is, and a basic result of integration theory (Riemann or Lebesgue) is that for any function $f$, $\int_a^a f(x)\,dx = 0$.

 

[independentphysics]

You seem confused. There is no arbitrary constant in a definite integral, which is what $M(R) = 2\pi\int_0^R r\rho(r)\,dr$ is, and a basic result of integration theory (Riemann or Lebesgue) is that for any function $f$, $\int_a^a f(x)\,dx = 0$.

Cumulative mass in function of r is not a definite integral.

 

[pasmith]

Cumulative mass in function of r is not a definite integral.

The limits are specified: the lower limit is fixed at zero, and the upper limit is the radius we are considering. That makes it a definite integral.

 

[SammyS]

There is a constant (+C) after integrating the density function to obtain cumulative mass M(r), and M(0) is not 0.

Yes:
You can use the indefinite integral to obtain the correct expression for the cumulative mass, M(r), of the disk. However, it makes sense that M(0) = 0 . That does not give you a value of 0 for the integration constant, C.

In Post#17, you had:

M (r)=2π * ρ0 * h * (h + r) * (-e^(-r/h)) + c
I find c by M(0)=c

Writing the LaTeX version of that, we have: $\displaystyle \quad M(r)=2\pi \rho_0 (h^2 + rh) (-e^{-r/h}) + C$

Giving that $\displaystyle \quad M(0)=C-2\pi \rho_0 h^2$.

Setting this to zero gives $\displaystyle \quad 2\pi \rho_0 h^2 =C$

Substitute that back into the expression for $M(r)$, and after a little rearranging, you get:

$\displaystyle \quad M(r)=2\pi \rho_0 \left(h^2(1-e^{-r/h}) - rh\,e^{-r/h} \right)$

 

[independentphysics]

Yes:
You can use the indefinite integral to obtain the correct expression for the cumulative mass, M(r), of the disk. However, it makes sense that M(0) = 0 . That does not give you a value of 0 for the integration constant, C.

In Post#17, you had:

Writing the LaTeX version of that, we have: $\displaystyle \quad M(r)=2\pi \rho_0 (h^2 + rh) (-e^{-r/h}) + C$

Giving that $\displaystyle \quad M(0)=C-2\pi \rho_0 h^2$.

Setting this to zero gives $\displaystyle \quad 2\pi \rho_0 h^2 =C$

Substitute that back into the expression for $M(r)$, and after a little rearranging, you get:

$\displaystyle \quad M(r)=2\pi \rho_0 \left(h^2(1-e^{-r/h}) - rh\,e^{-r/h} \right)$

I was wrong, I find this correct. Thank you very much