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

Explain about refracting telescopes, Refracting telescopes, if uncorrected,...

Refracting telescopes, if uncorrected, suffer from chromatic aberration.  Chromatic aberration occurs because: a)  red light is focused more strongly than blue light b)

Uses of radio isotopes, Uses of Radio Isotopes: Industries: 1. Radio...

Uses of Radio Isotopes: Industries: 1. Radio active isotopes are used to detect leakage in pipes. 2. It estimates weak and tear machine tools. 3. It is used to check t

Energy levels, ENERGY LEVELS The energy levels, measured in electron vo...

ENERGY LEVELS The energy levels, measured in electron volts (e.v.) in which orbiting electrons exist comply with a law of physics which states that energy can be given to elect

Illustrate tangent law in magnetism, Illustrate "Tangent law" in magnetism....

Illustrate "Tangent law" in magnetism. Compare magnetic moment of two bar magnet in a tan A, tan B position by null method or equal distance method.

A sodium cation has different number of electrons from, A sodium cation has...

A sodium cation has different number of electrons from: (1) O 2-                  (2) F -       (3) Li +                    (4) Al +3 Ans: Li +

Magnitudes, if A=[30.1, 21.4 degrees] and B= [25, 40 degrees] what is the m...

if A=[30.1, 21.4 degrees] and B= [25, 40 degrees] what is the magnitude of the sum of these two vectors

Explain displacement, DISPLACEMENT: The change in the position of a bod...

DISPLACEMENT: The change in the position of a body from its starting position to its final position is known as displacement. It is denoted by d ?.

Calculate the absolute pressure within the chamber in bars, (i) A closed sy...

(i) A closed system consisting of 2 kg of a gas undergoes a process during which the relationship between pressure and specific volume is pv1.3 = c. The process begins with 1 p =1b

., What is plane diffraction grating? How the wavelength of sodium light is...

What is plane diffraction grating? How the wavelength of sodium light is measured?question #Minimum 100 words accepted#

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