Gravitational field inside a spherical shell, Physics

Assignment Help:

Gravitational Field inside a Spherical Shell

In this problem we prove that the gravitational field inside a thin spherical shell of finitemass is zero. By extension, the field inside a thick shellwhoseinner and outer radii are finite is also zero. In this gure, R is the radius ofthe shell andMits mass. The mass per unit area is σ = M/S where S is the surface area. An object of mass m is located at point p, which is at distancer from the center of the sphere.

An element of mass is given by dM = σ × 2π R × R sin θ dθ, which corresponds to the ring on the surface. If the gravitational force along the lines is resolved into inline and perpendicular components, we find that the perpendicular components (Rsin θ) cancel and only the inline (Rcos θ) components contribute. Therefore the acceleration due to a gravitational element is:

da =GdM/s2cos φ.

2147_Gravitational Field inside a Spherical Shell.png

Substituting for dM as well as integrating over the surface

We have

a = σ2πG∫πθ=0 (cos φ sin θ/s2)dθ.

Currently we express s and φ in terms of θ.

s2= R2+ r2- 2R r cos θ

Using the law of cosines. Differentiating,                         

1437_Gravitational Field inside a Spherical Shell1.png

We are able to now make the necessary substitutions, mindful that changing the variable of integration changes the limits.

1582_Gravitational Field inside a Spherical Shell2.png

Now σ = M/4π R2 consequently

1244_Gravitational Field inside a Spherical Shell3.png

Integrating,

1805_Gravitational Field inside a Spherical Shell4.png

Thus the acceleration due to gravity at any point inside a thin spherical shell is identically zero! There are other simpler approaches to find this solution. For example Gauss' Law immediately yields the same result.


Related Discussions:- Gravitational field inside a spherical shell

Determine the radius of curvature of glass sphere, A small air bubble is pl...

A small air bubble is placed 0.04 m inside a glass sphere of refractive index 1.5. When viewed normally from outside it is seen at a depth of 0.03 m. Determine the radius of curvat

Calculate the sunrise and sunset local horizon points, This problem examine...

This problem examines the seasonal solar radiation budget at three geographical locations: the Equator, 40º North, and 80º North. NOTES: At a given geographical location, N

Electromagnetic waves, Electromagnetic waves: Electromagnetic waves wi...

Electromagnetic waves: Electromagnetic waves with different ranges of frequency constitute an electromagnetic spectrum. Electromagnetic waves are transverse in nature. Visible

Bomb Calorimetry, What is the difference between an adiabatic and an isoper...

What is the difference between an adiabatic and an isoperibol bomb calorimeter?

What are three dimensional wave , Energy in transmitted in space in all dir...

Energy in transmitted in space in all direction. Example : Light and sound waves propagating in space.

What are the stars, What are The Stars? The star is made of dust and ga...

What are The Stars? The star is made of dust and gas in space. Essentially it has many stages they are 1.black hole 2.nuetron star 3.supernova 4.red giant stage

What is a fiber optic system, What is a fiber optic system? This term c...

What is a fiber optic system? This term comprises all the essential elements to install a working unit.

What is the direction of the velocity after the collision, A 50.0-g toy car...

A 50.0-g toy car travelling with a velocity of 3.00 m/s due north collides head-on with a 180.0-g fire truck travelling with a velocity of 0.50 m/s due south. The toys stick to

Classical mechanics., what is virtual displacement?and how is it different ...

what is virtual displacement?and how is it different from actual displacement?

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