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

How many neutrons are in nucleus, An isotope of iodine (Z = 53) is used to ...

An isotope of iodine (Z = 53) is used to treat thyroid conditions. Its mass number is 131. How many neutrons are in its nucleus?   131 - 53 = 78 neutrons

Determine the maximum amount of power, Water accelerated by a nozzle to 20 ...

Water accelerated by a nozzle to 20 m/s strikes the vertical back surface of a cart moving horizontally at a constant velocity of 5 m/s in the flow direction. The mass flow rate of

Calculate the temperature after compression, A substance has constant heat ...

A substance has constant heat capacity C P = 3.5R and its equation of state is given by: P (V-cRT) = RT where c = 5 * 10 -7 m 3 /J. This substance is isentropically compre

Coordinate system for circular motion, Coordinate System for Circular Motio...

Coordinate System for Circular Motion In physics and circular motion is a movement of an object that was along the circumference of a circle or rotation along a circular path.

Problems on digital image, What is the raw file size for an image of 80 row...

What is the raw file size for an image of 80 rows and 100 columns, with pixels on 14 bits and with an extra row for the header? The minimum and maximum values observed in a

Determine the voltage gain, If the input voltage for the op amp shown in Fi...

If the input voltage for the op amp shown in Figure, is -0.5V, Determine (a) The voltage gain (b) The output voltage.

Determine the refractive index of glass, A glass sphere of radius 15 cm has...

A glass sphere of radius 15 cm has a small air bubble at a distance of 6 cm from its centre. When viewed normally along its diameter from the nearest side, the bubble seems to be 7

Can multiple illuminators change colors simultaneously, Can multiple illumi...

Can multiple illuminators change colors simultaneously? Yes. Usually the standard disc rotating motors are of the synchronous form, very reliable, and geared to the most suffi

Electrical scientists-discover the atom research paper, how would you descr...

how would you describe the discoveries of Benjamin franklin, Robert millikan, cgarles-augastin de coulomb and Ernest ruthford?

Two long wires, a) Two long wires, carrying currents i 1 and i 2 , are pla...

a) Two long wires, carrying currents i 1 and i 2 , are placed perpendicular to each other in such a way that they just avoid to contact. Determine the magnetic force on a small le

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