How Long Does It Take for a Particle to Travel Through an Earth-Crossing Tunnel?

[Avi1995]

Problem: Assume that a tunnel is dug across the earth(radius=R) passing through its centre. Find the time a particle takes to cover the length of tunnel if it is projected into the tunnel with a speed of $\sqrt{gR}$
2. Homework Equations :
Basic SHM equations:
1.F=-kx
2.T=(2*pi)/(omega)
Gravitation:
$F=GMm/r^2$
Attempt:
V(potential due to earth)=GMx/R(R+x)
1/2*m*g*R=GMx/R(R+x)
x=R
So particle will first go inside the tunnel accelerating till centre then decelerating till surface and further till it reaches distance R from surface.
I am confused now As this is not a SHM(Outside surface the field is non linear) So how do I find the time?

 

[deep838]

g`=GMx/R3
Thats the acceleration at a distance x from the center.
So the acceleration of the particle is directly proportional to the (-ve) of the distance from the center.
So the particle will execute a Simple Harmonic Motion with its circular frequency, ω²=GM/R3
Now could you work it out?

 

[Avi1995]

But outside the surface the field is non linear, so it won't be a shm.

 

[deep838]

But outside the surface the field is non linear, so it won't be a shm.

The particle won't be able to go outside! The force of gravity will keep on increasing just enought to make its velocity 0 at the surface! Think about it

 

[Avi1995]

The particle won't be able to go outside! The force of gravity will keep on increasing just enought to make its velocity 0 at the surface! Think about it

But there is no change in potential energy from one end to other end as both are on surface. So shouldn't K.E. remain same?

 

[ehild]

Read the problem carefully: it asks how long the particle is inside the tunnel.

In the tunnel, the particle performs SHM. Why? What force acts on the particle at distance r<R from the centre of Earth? Assume that the density of the Earth is constant. You have to know that the force exerted by a homogeneous sphere is the same as if all mass enclosed in the sphere of radius r concentrated in the centre.

ehild

 

[haruspex]

g`=GMx/R3
Thats the acceleration at a distance x from the center.

No. At radius r from the centre of a solid sphere radius R>r (assuming each concentric shell is in itself uniform), the gravitational pull from the portion of the sphere at radius > r exactly cancels itself. So only consider the pull from the part of the Earth at radius < r.

 

[deep838]

No. At radius r from the centre of a solid sphere radius R>r (assuming each concentric shell is in itself uniform), the gravitational pull from the portion of the sphere at radius > r exactly cancels itself. So only consider the pull from the part of the Earth at radius < r.

yeah that's what i did to get the equation!

 

[haruspex]

yeah that's what i did to get the equation!

Sorry - misread it. Too hasty.

 

[Avi1995]

Read the problem carefully: it asks how long the particle is inside the tunnel.

In the tunnel, the particle performs SHM. Why? What force acts on the particle at distance r<R from the centre of Earth? Assume that the density of the Earth is constant. You have to know that the force exerted by a homogeneous sphere is the same as if all mass enclosed in the sphere of radius r concentrated in the centre.

ehild

Understood. But the little doubt that remains in my mind is that particle was projected with a speed in the tunnel, can the equations of SHM be used?

 

[ehild]

The initial velocity and initial position determines the amplitude and phase constant of the SHM, but the equation ma=-kx does not change.

ehild