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

Rotation, A bicycle has tires of radius 0.35 meters. If the bicycle is tra...

A bicycle has tires of radius 0.35 meters. If the bicycle is traveling at a constant speed of 7.8 m/s, at approximately what angular speed are the tires rotating?

What is a mole? , It is the molecular weight of a substance expressed in gr...

It is the molecular weight of a substance expressed in grams ; the basic unit of amount of substance adopted under the Systeme International d''Unites.

Describe the theory of black body, Describe the theory of black body. B...

Describe the theory of black body. Black body At an exact temperature the black body would produce the maximum amount of energy probable for temperature. This would emit at

Ferromgntic substances, #question.An experminet to prove that a substance i...

#question.An experminet to prove that a substance is ferrmoanetic

Satellite motion, Satellite Motion A satellite is acted upon through th...

Satellite Motion A satellite is acted upon through the force of gravity and this force does accelerate it towards the Earth. Within the absence of gravity a satellite would mov

New potential difference, A parallel plate capacitor has plates of area 200...

A parallel plate capacitor has plates of area 200 cm 2 and separation among the plates 1.00 mm. What potential difference will be manufactured if the charge of 1 nC is given to th

Explain squirrel cage induction motors, Mines Flame proof squirrel cage...

Mines Flame proof squirrel cage induction motors are used in coal mine.

Polystyrene film capacitors, Polystyrene film is extruded flat and free fro...

Polystyrene film is extruded flat and free from pin-holes. Film is made from polystyrene granules by extruding heated tube of polymer which is then spread out and stretched state.

Electric field., why do electric field lines contract longitudinally

why do electric field lines contract longitudinally

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