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

Determine the current flowing in the conductor, A 25cm long conductor moves...

A 25cm long conductor moves at a uniform speed of 8m/s by a uniform magnetic field of flux density 1.2T.  Verify the current flowing in the conductor when (a) Its ends are o

Find the effectual resistance between a and b, Q. Find the effectual resist...

Q. Find the effectual resistance between A and B. 1/R p = 1/R 1 + 1/R 2 R p = R 1 R 2 / (R 1 + R 2 )          = 15 X15 / 30 = 7.5 ohm

Explain electrical field, Normal 0 false false false EN...

Normal 0 false false false EN-IN X-NONE X-NONE MicrosoftInternetExplorer4 A positive charge or

Determine element that atomic weight is double atomic number, The atomic we...

The atomic weight of an element is double its atomic number. If there are four electrons in 2P  orbital, the element is : (1) C                       (2) N       (3) O

Evaluate kinetic energy and momentum, If a proton and an electron have same...

If a proton and an electron have same de Broglie wavelength, which has more kinetic energy and momentum?

Determine the length of this heat exchanger, A concentric tube heat exchang...

A concentric tube heat exchanger is comprised of a thin wall inner tube of 25.4 mm diameter and an outer tube of 47 mm diameter. The heat exchanger is used for cooling lubricating

What are the image height and orientation, A 5.0-cm-tall object is placed 1...

A 5.0-cm-tall object is placed 16 cm from a convex lens with a focal length of 8.4 cm. What are the image height and orientation?

Center of mass, how to find center of mass of an object?

how to find center of mass of an object?

How much temperature copper constantan is used for measuring, Copper consta...

Copper constantan is used for measuring temperature up to (A) 1400 o C (B) 800 o C. (C) 1100 o C. (D) 400 o C.  Ans: Copper constantan is used for meas

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