Find the transit time of the train

[GLD223]

Homework Statement

The train will use gravity to traverse from point A to B, in your answer, use the gravitational constant g and R - the radius of the earth in your answer. Assume the earth's mass is uniform

Relevant Equations

$L = T-V, KE(0) = 0, PE(0) = KE + PE$

I need to find the transit time of the train, from what I see online it should be independent of points A and B and should be roughly 42 minutes.
I started by looking at the energies and derived an ODE for the speed of the train to find the transit time but it seems to be wrong.
Also, I might need to use the Lagrangian since we learned it in the last lesson.
$M_earth := M$
$\rho = M/4/3 * \pi * R^3$
$M(r) = r^3/R^3 * M$
$T = 1/2 * m *(dx/dt)^2$
$U(0) = -mgR$
$U(r) = -mg/R *r^2$

 

[Delta2]

Wikipedia has a quite good treatment on this, both explaining why the transit time is independent of the end points and finding the formula for the time, but wiki does it in the Newtonian formalism , that is with force and acceleration concepts and not with Lagrangian.

https://en.wikipedia.org/wiki/Gravity_train

 

[PeroK]

Generating a differential should work. You should rotate the diagram of the Earth so that the line AB is horizontal. It's seems silly not to do that. Try finding the equation of motion for the horizontal motion.

Perhaps using the Lagrangian is a shortcut.

 

[PeroK]

Wikipedia has a quite good treatment on this, both explaining why the transit time is independent of the end points and finding the formula for the time, but wiki does it in the Newtonian formalism , that is with force and acceleration concepts and not with Lagrangian.

https://en.wikipedia.org/wiki/Gravity_train

That's just looking at a solution, rather than trying to work it out for yourself.

 

[Delta2]

That's just looking at a solution, rather than trying to work it out for yourself.

Ehm, yes that's right but still you know how wiki explanations are sometimes, they mention step A and step H and they require for you to fill in the steps B,C,D,E,F,G, so the OP might need to work through wiki solution so he will fully understand it and fill in the in between steps by himself.

 

[PeroK]

Homework Statement: The train will use gravity to traverse from point A to B, in your answer, use the gravitational constant g and R - the radius of the earth in your answer. Assume the earth's mass is uniform
Relevant Equations: $L = T-V, KE(0) = 0, PE(0) = KE + PE$

I need to find the transit time of the train, from what I see online it should be independent of points A and B and should be roughly 42 minutes.
I started by looking at the energies and derived an ODE for the speed of the train to find the transit time but it seems to be wrong.
Also, I might need to use the Lagrangian since we learned it in the last lesson.
$M_earth := M$
$\rho = M/4/3 * \pi * R^3$
$M(r) = r^3/R^3 * M$
$T = 1/2 * m *(dx/dt)^2$
$U(0) = -mgR$
$U(r) = -mg/R *r^2$

The Lagrangian approach works quite well. But, you must be careful calculating the GPE inside a solid sphere. Your expression for $U(r)$ is not correct.

 

[GLD223]

The Lagrangian approach works quite well. But, you must be careful calculating the GPE inside a solid sphere. Your expression for $U(r)$ is not correct.

my expression for U is $-mgr^2/R$ that is what I meant (to multiply by r squared and not divide by r squared)

 

[PeroK]

my expression for U is $-mgr^2/R$ that is what I meant (to multiply by r squared and not divide by r squared)

That's not correct.

 

[GLD223]

That's not correct.

why not? The gravitational potential energy is given by $-GMm/r$ and $M(r) = (r^3/R^3) * M$.
Plugging this in and using $GM/R^2 = g$ I get $(-mg*r^2)/R$ which also makes sense if you plug in r=R (you get $-mgR$)

 

[PeroK]

why not? The gravitational potential energy is given by $-GMm/r$ and $M(r) = (r^3/R^3) * M$.
Plugging this in and using $GM/R^2 = g$ I get $(-mg*r^2)/R$ which also makes sense if you plug in r=R (you get $-mgR$)

Please calculate the gravitational force, as a function of $r$, based on your potential.

 

[GLD223]

Oh I think it needs to be $U(r) = -mgr^2 / 2R$
Because the force is $G*M(r)*m/r^2 = mgr/R$ and integrating this with a minus sign gives $-mgr^2 / 2R$ but I still get a wrong solution

 

[GLD223]

Please calculate the gravitational force, as a function of $r$, based on your potential.

$F = -d/dr(-mgr^2/R) = 2mgr/R$

 

[PeroK]

Oh I think it needs to be $U(r) = -mgr^2 / 2R$
Because the force is $G*M(r)*m/r^2 = mgr/R$ and integrating this with a minus sign gives $-mgr^2 / 2R$ but I still get a wrong solution

The sign is wrong. The force has a negative sign in the first place.

 

[PeroK]

$F = -d/dr(-mgr^2/R) = 2mgr/R$

Which is wrong. You should have started with the force and derived the potential from that.

 

[GLD223]

The sign is wrong. The force has a negative sign in the first place.

So $F = -G*M(r)*m/r^2$?

 

[PeroK]

So $F = -G*M(r)*m/r^2$?

Yes. You can get the potential from that.

 

[GLD223]

I start by deriving the potential given the aforementioned force $F=-G*M(r)*m/r^2 = -mgr/R => U(r) = mgr^2/2R$
Now there is no kinetic energy at time 0 so the total energy at time zero is $E(0) = U(r=R) = mgR/2$
From conservation of energy:
$mgR/2 = m*v^2/2 + mgr^2/2R$
now I can use the Pythagoras theorem to show that $r^2 = r_0^2 + (D/2 - x)^2$ problem is I don't have an easy way to express r_0 (I can use trigonometry but it gets messy) and after I figure out an expression for r using x I can generate an ODE for dx/dt (v) and find the transit time. How can I continue from here? I'm stuck

 

[PeroK]

I start by deriving the potential given the aforementioned force $F=-G*M(r)*m/r^2 = -mgr/R => U(r) = mgr^2/2R$
Now there is no kinetic energy at time 0 so the total energy at time zero is $E(0) = U(r=R) = mgR/2$
From conservation of energy:
$mgR/2 = m*v^2/2 + mgr^2/2R$
now I can use the Pythagoras theorem to show that $r^2 = r_0^2 + (D/2 - x)^2$ problem is I don't have an easy way to express r_0 (I can use trigonometry but it gets messy) and after I figure out an expression for r using x I can generate an ODE for dx/dt (v) and find the transit time. How can I continue from here? I'm stuck

I thought you were going for the Lagrangian approach?

In any case, you could try looking for the equation of motion for the horizontal displacement. The trig is fairly straightforward.

 

[GLD223]

update: using trigonometry worked but it is kinda dirty (sin of arccos) but it gave me the correct expression

 

[GLD223]

I thought you were going for the Lagrangian approach?

Yes I wanted to do it using conservation of energy first and now I'll do it using the Lagrangian approach

 

[GLD223]

@PeroK Thanks

 

[GLD223]

is there a nicer way to express r^2 using x? because I get $r^2 = r_0^2 + (D/2 - x)^2$ and $r_0 = R*sin(arccos(D/(2*R)))$

 

[PeroK]

is there a nicer way to express r^2 using x? because I get $r^2 = r_0^2 + (D/2 - x)^2$ and $r_0 = R*sin(arccos(D/(2*R)))$

I just expressed everything in terms of $x$.

 

[GLD223]

is there a nicer way to express r^2 using x? because I get $r^2 = r_0^2 + (D/2 - x)^2$ and $r_0 = R*sin(arccos(D/(2*R)))$

I asked the question too quickly but it doesn't matter since r_0 is still a constant and thus "disappears" in the E.L. equation