Tunnels through the earth, Physics

Assignment Help:

Q. Tunnels through the Earth

Assume a straight tunnel is cut from the surface of the earth to the opposite side through the centre.

The object is going down the tunnel and assuming that air friction is negligible we wish to determine the time it takes for the object to return to its starting point.

The only force acting on the object is gravitational. From previous problems we know that only that portion of the earth's mass contained in the sphere with radius equal to the object's height above the centre will contribute.

Let.

                                    σ =M/V

And.

                                    V =4/3π R3

Where M is the mass of the earth and V is the volume. Be the mass density of the earth. Pretentious uniform density at any radius from the center

σ =Mr/Vr.

The foremost equation for an object at any distance r from the center is

577_Tunnels through the Earth.png

The force, F is

F = mar =4/3πGσmr.

However this corresponds to the differential equation

d2r/dt2- k r = 0

Whose solution is the same as that of an object subject to a spring force without friction, which is simple harmonic motion the period is given by

2482_Tunnels through the Earth1.png

At the surface a gravitational force is the weight of the object

1459_Tunnels through the Earth2.png

Thus

g/R=4/3π σG,

so

      1045_Tunnels through the Earth3.png


Related Discussions:- Tunnels through the earth

What is the electric potential in hollow conducting shell, Five coulombs of...

Five coulombs of charge are placed on a thin-walled conducting shell. Once the charge has come to rest, the electric potential in the hollow conducting shell is found to be:

Evaluate the magnitude and direction of the total momentum, Two arrows are ...

Two arrows are fired horizontally with the same speed of25.0 m/s. Each arrow has a mass of0.116 kg. One is fired due east and the other due south. Find the magnitude and direction

An example of change in the momentum, A girl throws a 0.1 kilogram ball at ...

A girl throws a 0.1 kilogram ball at a wall.  The ball hits the wall perpendicularly with a velocity of 5 meters per second and then bounces straight back with a velocity of 4 mete

Ac output resistance in common emitter configuration, AC Output Resistance ...

AC Output Resistance in Common Emitter Configuration Ratio  of  change  in collector-emitter voltage (?V CE ) to the change in collector current (?I C ) at constant base curren

What is the path of the force exerted on the particle, A positively charged...

A positively charged particle is pitiful southward in a downward-directed uniform magnetic field. The magnetic force is the merely force if any acting on the particle. What is the

Vibrational modes of carbon dioxide molecule representation, How are vibrat...

How are vibrational modes of carbon dioxide molecule represented? Generally the modes are represented through three integers (mnq): a. (moo) represents Symmetric stretching

Nuclear reactions, estimation of nuclear fusion on basis of liquid drop mod...

estimation of nuclear fusion on basis of liquid drop model

Frequency dependence of dielectric constant, E lectrons have very much smal...

E lectrons have very much smaller mass than that of ions, so they respond more rapidly to a changing electric field. For electric field that oscillates at extremely high frequencie

Write Your Message!

Captcha
Free Assignment Quote

Assured A++ Grade

Get guaranteed satisfaction & time on delivery in every assignment order you paid with us! We ensure premium quality solution document along with free turntin report!

All rights reserved! Copyrights ©2019-2020 ExpertsMind IT Educational Pvt Ltd