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

Electricity, help by post offoce box explan rasistance of a wire

help by post offoce box explan rasistance of a wire

Voltage dependent resistor, A VDR is a special type of resistor that chang...

A VDR is a special type of resistor that changes its resistance with rise in voltage. It has a very high resistance at low voltage (below the trigger voltage) and very low resista

Find the temperature of the drink, (a) Two 40 g ice cubes are dropped into ...

(a) Two 40 g ice cubes are dropped into 220 g of water in a thermally insulated container. If the water was initially at25°C, and the ice came directly from a freezer at -15°C,what

Explain change in kinetic energy, Two blocks are tied together by a horizon...

Two blocks are tied together by a horizontal string and pulled a distance of 2.7 m across an air hockey table with a constant force of 35 N. The force is directed at an upward angl

How to make a sun-dial, Making a sun-dial To withstand all weather, a s...

Making a sun-dial To withstand all weather, a sun-dial should be form of metal or painted wood. A cardboard model can be formed for easy experiments.  The gnomon which casts

Explain selection of motors, Selection of motors for different Engg. applic...

Selection of motors for different Engg. application 1.  For Cement industry: Various types of motors for pumps, fans, conveyors ,crushers, kilns an mills Major mill

Applications of neodymium-yttrium-aluminium-garnet laser, What are the appl...

What are the applications of Neodymium-Yttrium-Aluminium-Garnet Laser? Applications of Neodymium-Yttrium-Aluminium-Garnet Laser: a. Used into materials processing as like we

What is the magnitude of the magnetic induction, The magnetic vector potent...

The magnetic vector potential in a specific region is a constant vector having magnitude 4πX10 5 units and points along the positive x-axis. What is the magnitude of the magnetic

Define perfectly plastic body , An object which does not get its actual con...

An object which does not get its actual configuration at all on the removal of deforming force, though short the deforming force can be is a perfectly plastic body.

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