Gravity problem with non constant acceleration

[inferno298]

Homework Statement

You are a space traveler and you land on the surface of a new planet. You decide to drill a tunnel through the center of the planet to the opposite side of the planet. Your idea is to use this tunnel to easily transport rocks from your location to the opposite side of the planet. So, you drop a rock in the tunnel and it falls down the hole toward the center of the planet, passes through the center of the planet, then makes it to the opposite side of the planet. Assume the planet is a uniform solid sphere of radius 8.50×10^6 m. The rock is dropped from rest into the hole, and it initially has an acceleration of 10.70 m/s2. How long will it take to make it to the other side of the planet?

Homework Equations

F = (G m1 m2)/r^2
U = -(G m1 m2)/r

The Attempt at a Solution

So I guess I am having a hard time starting this, which in essence is pretty much the only hard part of any physics problem. I pretty sure this will be a differential equation since the acceleration due to gravity will be changing as the object gets closer to the center of the sphere. I also believe that this might be able to be solved as a simple harmonic oscillator diff'eq.

First ill try and get the mass of the planet. Using a and r I can solve for the escape velocity and then use that to solve for the mass. Using equations:
Ve = Sqrt[2*a*r]
M = (Ve^2 * R)/(2*G)

Now F = m1 a
m1 = mass of stone
m1*a = (G m1 m2)/r^2

m1 cancels on both sides
a = dv/dt
dv/dt = (G m2)/r^2

so a will change but so will r, that's two changing variables since r is a function of t also. Would I have to do some Lagrange Equations of Motions? This where I am lost at. Don't know if I am making this more complicated then it needs to be.

 

[nasu]

Where did that escape velocity came from? I don't see it mentioned in the problem.
You could find the mass of the planet by using the value of the surface acceleration of gravity but the answer does not depend on the mass so finding the mass is not really useful.

When you write the force of gravity on the m1, what is the significance of m2?

 

[inferno298]

it was just a round about way to get the mass of the planet, escape velocity didn't need to be mentioned. All the tools were there. I just forgot the direct way to get to the mass. As for m2, that is the mass of the planet, m1 cancels out. m2 didn't cancel out, therefore it was still part of the Gravitational central force equation.

Any more help on how to get going would be appreciated

 

[nasu]

m2 should not be the entire mass of the planet.
When the body is at distance r from the center (in that tunnel), with r<R, only the mass of the sphere of radius r enters the formula for gravitational attraction. Here R is the radius of the planet.

 

[Nickie]

You are correct in thinking that this problem can be solved using differential equations. The changing acceleration due to gravity as the rock moves closer to the center of the planet makes this a non-constant acceleration problem. To solve this, you can use the equation for Newton's second law, F=ma, where the force is given by the gravitational force between the rock and the planet, and the acceleration is the rate of change of velocity.

You can start by setting up a differential equation for the acceleration of the rock as it moves through the planet. This can be done using the equation for the gravitational force, F=(Gm1m2)/r^2, and the equation for acceleration, a=dv/dt. You will need to use the mass of the planet, which you have already calculated, and the mass of the rock, which is given in the problem.

Once you have set up the differential equation, you can solve it using techniques such as separation of variables or variation of parameters. This will give you the velocity of the rock as a function of time. Then, you can integrate the velocity function over time to find the distance traveled by the rock.

Alternatively, as you mentioned, this problem can also be solved using Lagrange's equations of motion. In this method, you will need to define a Lagrangian function for the system, which takes into account the kinetic and potential energies of the rock as it moves through the planet. Then, you can use the Euler-Lagrange equations to solve for the motion of the rock.

Both methods will give you the time it takes for the rock to reach the other side of the planet. I hope this helps you in solving the problem. Remember to always start with the basic principles and equations and break down the problem into smaller, more manageable parts. Good luck!