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

What are solid core fibers, What are solid core fibers? This optics is ...

What are solid core fibers? This optics is cylinders made of diverse polymers and encased upon a transparent tube or sheath. As a standard are manufactured in various gauges or

What is the frequency of the vibrator, In the arrangement shown in figure, ...

In the arrangement shown in figure, an object can be hung from a string (with linear mass density µ = 0.002 00 kg/m) that passes over a light pulley. The string is connected to a v

Bernoullis theorem, what is the theory behind bernoullies theorem?

what is the theory behind bernoullies theorem?

Friction, A thin rod of length 1 m is fixed in a vertical position inside a...

A thin rod of length 1 m is fixed in a vertical position inside a train, which is moving horizontally with constant acceleration 4 m/s2. A bead can slide on the rod, and friction

Elastic collision-car-truck collisions, Elastic Collision Collisions am...

Elastic Collision Collisions among objects are governed through laws of momentum and energy. While a collision occurs in an isolated system, the whole momentum of the system of

Hoop conjecture, Hoop conjecture (K.S. Thorne, 1972): The conjecture (...

Hoop conjecture (K.S. Thorne, 1972): The conjecture (as still unproven, though there is substantial evidence to hold up it) that a nonspherically compressed, nonspherical obje

A parallel plate capacitor each with plate area a, A parallel plate capacit...

A parallel plate capacitor, each with plate area A and separation d, is charged to a potential dissimilarity V. The battery used to charge it is then disconnected. A dielectric sla

How far will it travel in that time interval, If the velocity of the box in...

If the velocity of the box in problem 10 was 5.0 m/s at time zero, what will be its speed after 0.50 s? How far will it travel in that time interval?

Illustrate what are the weights in pounds of liquid gallons, Q. Illustrate ...

Q. Illustrate what are the weights in pounds of liquid gallons? Answer:- First you should know the specific gravity of the liquid then you can compute the weight by multip

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