Can You Solve for R in Terms of t Using This Integral?

  • Context: Graduate 
  • Thread starter Thread starter Odyssey
  • Start date Start date
  • Tags Tags
    Integration
Click For Summary

Discussion Overview

The discussion revolves around the integration of a differential equation related to the motion of a point mass under a specific force law, particularly focusing on how to express the radius R in terms of time t through an integral. The context includes concepts from classical mechanics, such as angular momentum and energy conservation, and explores the implications of a non-standard force law (inverse cubic) on orbital dynamics.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Initial confusion regarding the integral and the variables involved, particularly the definitions of θ, R, u, and t.
  • Some participants suggest that the integral relates to solids of revolution, while others express uncertainty about the limits of integration.
  • A participant proposes that the problem involves a separable differential equation, but the specifics remain unclear.
  • Discussion about the conservation of total energy and angular momentum in the context of a central force, leading to expressions for energy and angular momentum in polar coordinates.
  • There are multiple attempts to clarify the integral's form and the role of R as a function of θ, with some participants confirming the correct expression.
  • Concerns are raised about the integration process and the need to eliminate time dependence in favor of angular dependence.
  • Participants explore the implications of a 1/r^3 force law and its effects on orbital motion, with suggestions for how to approach the problem mathematically.
  • There are discussions about the potential energy function and its relationship to the force, with some confusion over signs and constants involved.
  • One participant expresses uncertainty about how to incorporate the potential energy into the integral and how to proceed with the integration.

Areas of Agreement / Disagreement

Participants generally agree on the need to clarify the integral and the definitions of the variables involved, but there is no consensus on the correct approach to solving for R in terms of t or the implications of the force law on the motion of the mass. Multiple competing views and uncertainties remain throughout the discussion.

Contextual Notes

There are limitations in the clarity of variable definitions and the assumptions underlying the integral. The discussion also reflects unresolved mathematical steps and the complexities of integrating under the specified force law.

Odyssey
Messages
84
Reaction score
0
How do I intergrate this?

[tex]t - to = \int_\Theta ^\Theta(t) \frac{m}{L} R(u)^2 du[/tex]

u is [tex]\Theta(t)[/tex], [tex]\Theta[/tex] is the angle at time zero, and t is time. Can I solve for R in terms of t?

Thanks.
 
Last edited:
Physics news on Phys.org
What you've written is completely incomprehensible.
What are "theta" and R?
What is u and t?
 
arildno said:
What you've written is completely incomprehensible.
What are "theta" and R?
What is u and t?

I think he is doing solids of revolution. that is the only thing that I can think of as to why there would be an R in there...but from theta to theta makes no sense to me.
 
Sorry for the confusion of the integral. I am new to Latex...and I don't know how to do the "integral from theta to theta(t)" (theta of t). I didn't mean theta to theta. =\
 
A good guess; I think it is some sort of separable differential equation he started up with, but I'm not sure..
 
Odyssey: Could you state the original problem?
 
arildno said:
Odyssey: Could you state the original problem?

Thanks for the help. I appreciate it! =D

OK, my problem is actually much more complicated than the integral.

Suppose a point mass m experiences a net force F = -Amr^-3.
How should I go about in describing its orbits for E>0, E <0, and E = 0 for the non-zero angular momentum cases? :confused:

My prof gave me an example on the gravitational force law, which is an inverse square law. I am asked to find out how the particle would move (say Earth) if it is under the influence of an inverse CUBIC law. The integral I (tried to) give above is the angular momentum of the particle. And yes, it is a differential equation! :)
 
Last edited:
Is this how it should be?
[tex]t-t_{0}=\int_{\theta_{0}}^{\theta(t)}\frac{m}{L}R^{2}(u)du[/tex]
 
arildno said:
Is this how it should be?
[tex]t-t_{0}=\int_{\theta_{0}}^{\theta(t)}\frac{m}{L}R^{2}(u)du[/tex]

Precisely! Yes, that's how it should be! :smile:


Hummmm, I did some research on the Net, and I *think* it wouldn't be very pleasant if we live in a 1/r^3 world. The Earth would spiral into the Sun??!? I am asked to find the equation of the orbit of such a world. Can you give me some pointers so I can start this problem please?

Thx for the help again.
 
Last edited:
  • #10
Allright!
So, am I correct in assuming that "R" is the unknown radius in some orbit expressed as a function of the angle?
 
  • #11
arildno said:
Allright!
So, am I correct in assuming that "R" is the unknown radius in some orbit expressed as a function of the angle?

Yes, precisely. R is a function of the angle, which is not limited to just between 0 and 2pi.

Let me get some more info on the problem. That might help! :) (will be posted here in minutes)
 
Last edited:
  • #12
Well, then you can't integrate any further.
If you don't know what you integrate, you can't know what the integral would be.
(That is, what you've got is a dead-end)
 
  • #13
Total energy is conserved for a conservative force.

E = (1/2) m [r'(t)]^2 + (1/2) m [r(t)]^2 [theta'(t)]^2 + V (r(t))

Angular momentum is conserved for a central force.

L = m [r(t)]^2 [theta'(t)]^2

we can view r may be a function of theta, assuming that L is non-zero.

then E = (1/2) m[R'(theta(t))]^2 [theta'(t)]^2 + (1/2) m [R(theta(t))]^2 [theta'(t)]^2 + V(R(theta(t)))
and
L = m [R(theta(t))]^2 [theta'(t)]
 
  • #14
Hummm, under a 1/r^3 force, and using angular momentum/total energy conservation (writting the equations in polar coordinate form too), I separate the variables theta and t, and try to integrate to solve for R in terms of t (in trying to get an equation of the orbit). Am i heading into the right direction? That's how I got the integral at the very top in the first place.
 
  • #15
True enough; try to eliminate your time dependence, (using angular momentum equation),
[tex]\frac{d\theta}{dt}=\frac{L}{mR^{2}}[/tex]

You should then by able to find a differential equation for R expressed in the new independent variable [tex]\theta[/tex]
(Basically, this is the same technique used to find the orbits for an inverse-square field)
Good luck!
 
  • #16
Note:
DO NOT KEEP t AS THE INDEPENDENT VARIABLE!
Switch to using the angle.
 
  • #17
OK, let me try to work this out...thanks for the help arildno! =D

I need the potential V(R) too. Just to verify the work. Is V(R) = Hmr^-2?
 
Last edited:
  • #18
If H is some constant related to your A, yes; then the potential goes as the inverse square of the radius.
 
  • #19
arildno said:
Is this how it should be?
[tex]t-t_{0}=\int_{\theta_{0}}^{\theta(t)}\frac{m}{L}R^{2}(u)du[/tex]

Actually I just found out the integral should be

[tex]t-t_{0}=\int_{\theta_{0}}^{\theta(t)}\frac{m}{L}R(u)^{2}du[/tex]

So do I still treat R as a constant? No?
 
  • #20
arildno said:
If H is some constant related to your A, yes; then the potential goes as the inverse square of the radius.

OOps! Sorry, I used two variables for the same constant! A = H! Silly me. :-p
 
  • #21
Careful!
There's a factor of "2" somewhere which you just forgot..
 
  • #22
arildno said:
Careful!
There's a factor of "2" somewhere which you just forgot..

In the potential? :confused: Let me try to get the energy equation to integrate.
 
  • #23
It's about getting from your F to your V
(In addition, be careful about the sign as well..)
 
  • #24
[tex]t-t_{0}=\int_{R_{0}}^{R(t)}\frac{du}{\sqrt{2mEL^{-2}u^4-u^2-2mL^{-2}u^4V(u)}}[/tex]

where, u = R of theta. So should I plug the V = -(1/2)Amr^-2 into V(u)? :confused:
 
Last edited:
  • #25
arildno said:
It's about getting from your F to your V
(In addition, be careful about the sign as well..)

Ah yes, (1/2)Amr^-2. The sign...it should be positive...?
 
  • #26
What's your argument?
 
  • #27
arildno said:
What's your argument?

Hummm, this is where I am stuck. I don't know how to proceed with the integral 3 posts above. :confused:

Integration of that integral would allow me to solve for R in terms of theta, which will in turn tell me the shape of the orbit of any test particle about the origin, in terms of its energy and angular momentum.

But the thing is, I don't know how to use the term [tex]V(u)[/tex] in the integral, where [tex]u = R(\Theta)[/tex]
 
Last edited:
  • #28
I meant why you think the sign should be positive.
 
  • #29
Is it true that [tex]V = -\int F[/tex]? =\
 
  • #30
Wait. If that's true then it should be negative. AHHHH! ><
 

Similar threads

Undergrad Having trouble understanding a seemingly simple u-substitution
  • · Replies 5 ·
Replies
5
Views
3K
High School Question about change of variables
  • · Replies 29 ·
Replies
29
Views
6K
Undergrad Integration Using Trigonometric Substitution
  • · Replies 8 ·
Replies
8
Views
3K
Undergrad Landscape Fabric Roll
  • · Replies 8 ·
Replies
8
Views
4K
High School Having trouble deriving the volume of an elliptical ring toroid
  • · Replies 4 ·
Replies
4
Views
5K
High School Question about the limits of integration in a change of variables
  • · Replies 2 ·
Replies
2
Views
4K
Undergrad Conservation of Mass for Compressible Flow
  • · Replies 33 ·
2
Replies
33
Views
4K
Graduate Function expansion in Cartesian and spherical tensors
  • · Replies 1 ·
Replies
1
Views
2K
Graduate How do I apply the method of steepest descents to this integral?
  • · Replies 1 ·
Replies
1
Views
3K
Undergrad On convolution theorem of Laplace transform: Schiff
  • · Replies 4 ·
Replies
4
Views
3K