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

heat transfer, Heat may be transmitted from one body to the other by any o...

Heat may be transmitted from one body to the other by any of three possible ways: conduction, radiation and convection.  In the conduction, convection phenomena, a medium is essent

Gravitation- numerical probelm, A ball dropped from a balloon at rest clear...

A ball dropped from a balloon at rest clears a tower  81m high during the last quater second of its journey. Find the height of the balloon and the velocity of the body when it rea

What are the merits of typical ultrasonic flaw detector, What are the merit...

What are the merits of typical ultrasonic flaw detector? Merits of typical ultrasonic flaw detector: a. Particularly one side of the specimen is needed for the study. b.

Relation between linear and angular variable, what is the relation between...

what is the relation between linear and angular variable?

What is joule cycle, What is Joule Cycle? Find the efficiency of the J...

What is Joule Cycle? Find the efficiency of the Joule cycle, consisting of two isobars and two adiabats.  Presume that the heat capacities of the gas C p and C v are constan

To find velocity of spacecraft and acceleration, find acceleration of space...

find acceleration of spacecraft will changed during journey and velocity and displacement if formula for displacement " s " and velocity V S=ut +1/2 at.... (t to power 2) V= u+at

Electric potential, is the surface of a conductor is equipotential

is the surface of a conductor is equipotential

Bcs theory, BCS theory (J. Bardeen, L.N. Cooper, J.R. Schrieffer; 1957) ...

BCS theory (J. Bardeen, L.N. Cooper, J.R. Schrieffer; 1957) A theory put forth to describe superconductivity and superfluidity both. This recommend that in the superconducting

Static limit, Static limit: The distance through a rotating black hole...

Static limit: The distance through a rotating black hole where no observer can probably remain at rest (w.r.t. the distant stars) due to inertial frame dragging; this region i

What is surface tension, It is found that when a liquid is free from the ex...

It is found that when a liquid is free from the external forces, it always gets the shape of a spherical drop. It is due to, for a needed mass, a sphere has the least field area. I

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