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

Which motor is used in textile industry, Textile industry: High speed d...

Textile industry: High speed drives are essential with wide range of control, small & medium motors are employed. Commutator motors and synchronous motors are used.

Full wave bridge, Explain full wave bridge circuits showing the waves

Explain full wave bridge circuits showing the waves

What is the kinetic energy of the block subsequent, A block of mass m on a ...

A block of mass m on a plane horizontal frictionless surface is pushed up against the end of a horizontal spring the other end of which is connected to a wall therefore that it com

Define neutron, Neutron is a fundamental particle which is necessary consti...

Neutron is a fundamental particle which is necessary constituent of all nuclei except that of hydrogen molecule. It was discovered by Chadwick. (1) The charge of neutron: It is

Determine the x and y coordinates - vertical circle, A 0.500 kg stone attac...

A 0.500 kg stone attached to the end of a string is whirled in a vertical circle of radius 1,20 meters. The center of the vertical circle is 1.50 meters above the ground. When the

Explain newton’s law of cooling , It is described to this law, if the tempe...

It is described to this law, if the temperature T of the object is not very dissimilar from that of the environment T0, then rate of cooling -dT/dt is proportional to the temperatu

Mass energy relation, Until the time of Einstein, mass and energy were cons...

Until the time of Einstein, mass and energy were considered as two different physical quantities. But in 1905, with the help of postulates of the special theory of relativity Einst

Band theory.., differentiate between direct and indirect band gaps

differentiate between direct and indirect band gaps

Thomson''s effect, In Thomson's effect we pact with only metallic rod and n...

In Thomson's effect we pact with only metallic rod and not with thermocouple as in Peltiers effect and Seebeck's effect. (That's why sometimes it is called as homogeneous thermo el

Deformation in a real body, DEFORMATION IN A REAL BODY DUE TO APPLICATION O...

DEFORMATION IN A REAL BODY DUE TO APPLICATION OF FORCE The idea of a hard body, where the constituent molecules are fixed with respect to each other, is theoretical. Till how we

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