Variable acceleration problem.

[frankR]

A particle of mass m is released from rest at a distance b from a fixed origin of force that attracts the particle according to the inverse square law:

F = -kx^-2

Show that the time required for the particle to reach the origin is:

[pi](mb^3/8k)^1/2

I have no idea where the pi came from.

This is what I've done.

F=dp/dt
dp/dt=mdv/dt
dv/dt=d^2x/dt^2
m*d^2x/dt^2=m*dv/dt
m*d^2x/dt^2=-k/x^2
m*x^2*d^2x=-k*dt^2
m[inte][inte]x^2dxdx=-k[inte][inte]dtdt

After solving the double integrals and pluging in the constants I get.

t = sqrt[-1/(6k)*m(16b^4)]

I'm going to be really embarrased if my calculus is wrong.

So am I doing wrong. I still have no idea where a pi comes from!

Thanks

Frank

 

[quantumdude]

Originally posted by frankR
m*d^2x/dt^2=-k/x^2
m*x^2*d^2x=-k*dt^2

Here's your problem. d²x/dt² cannot be separated as dxdx/dtdt.

You'll have much better luck by noting the following:

a=dv/dt
a=(dv/dx)(dx/dt)
a=(dv/dx)(v)

or

a=v(dv/dx)

Now, try that with your force:

-k/x²=mv(dv/dx)
(-k/x²)dx=mvdv

You can get x in terms of v, and then simply note that v=dx/dt and integrate again to find the time. Give it a shot and let me know if you need more help.

edit: fixed an omission

 

[frankR]

I get -b*sqrt(2)/(3*sqrt(k/(b*m)) = sqrt(2*b^3*m/9k)

I still don't see where a pi could come from.

 

[quantumdude]

Originally posted by frankR
I get -b*sqrt(2)/(3*sqrt(k/(b*m)) = sqrt(2*b^3*m/9k)

I still don't see where a pi could come from.

Can you show your work? That will help a lot!

Here's how I started:

-(k/x²)dx=mvdv

(sorry, I accidentally left the 'm' out in my last post, but I have fixed it)

-k[inte]_b^x(1/x²)dx=m[inte]₀^vvdv
(k/x)-(k/b)=(1/2)mv²

Are you with me so far? If so, can you show me how you did the second integral?

 

[frankR]

Originally posted by Tom

(k/x)-(k/b)=(1/2)mv²

Except for up to there. I made an algebriac mistake.

k/x - k/b != k/(x-b)

Let me start over. Give me a couple minutes.

Thanks for your help so far.

 

[frankR]

Okay cool.

The integral ins't too prety so I let me Ti-89 solve it.

I get t = -Sin([oo])b^3/2*sqrt(2m)[pi]/4sqrt(k)

Which is = [pi]sqrt[Sin([oo])*b^3m/8k]

What the heck does Sin([oo]) mean?

Thanks. I just need to slow down. Stop making dumb mistakes!

I need to stop forgetting that chain rule property.

Thanks a lot!

Frank

 

[quantumdude]

Originally posted by frankR
What the heck does Sin([oo]) mean?

Uh-oh.

It sin([oo]) is indeterminate, because it keeps oscillating. That means you can't assign a value to it. Me smells a mistake here.

 

[frankR]

Originally posted by Tom
Uh-oh.

It sin([oo]) is indeterminate, because it keeps oscillating.

I'm it's the software in my calculator.

I get a headache looking at this integral:

sqrt[m/2][inte]dx/sqrt[k/x-k/b]

I'm sure it's not as hard as it looks. I can drag up my old Calculus text, or look for a tabulated solution, to solve it another way.

 

[quantumdude]

I would try to work it into a form that matches an integral in a table. My guess is that it will turn out to be an inverse trig function, which is how you will get a π out of it.

 

[frankR]

Originally posted by Tom
I would try to work it into a form that matches an integral in a table. My guess is that it will turn out to be an inverse trig function, which is how you will get a π out of it.

I've been playing with it for half an hour, I'm not getting anywhere. It's beyond my mathematical abilities. Everything I do I still get an indeterminate solution.

If b>0, which it is, then I believe we get a solution.

 

[quantumdude]

Originally posted by frankR
I've been playing with it for half an hour, I'm not getting anywhere. It's beyond my mathematical abilities. Everything I do I still get an indeterminate solution.

Without knowing what you've done, I really can't say much on this.

If b>0, which it is, then I believe we get a solution.

Physically, you should get a solution, because this force is the same as a point mass or a point charge. Mathematically however, there seems to be a problem, because you encounter an integral like this:

[inte]_b⁰((x/(b-x))^1/2dx

whose integrand diverges at the lower boundary. Still, there should be a way to handle this improper integral.

 

[frankR]

Originally posted by Tom
[inte]_b⁰((x/(b-x))^1/2dx

That's the form I found.

If x is under the following condition:

0<=x<b

The solution is fine.