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**Gravitational Fields 1**

Newton's Law of gravitation |

__Newton's Law of Gravitation__

**For two masses displaced a distance apart, the gravitational force of attraction of one**

** mass on the other is directly proportional to the product of the two masses and**

** inversely proportional to the square of the distance between them. **

If the masses are *m*_{1} and** m**_{2} , with their centres of mass displaced a distance * r* apart, then the force of attraction

*of one mass on the other is described as:*

**F**

The proportionality can be made into an equation using a constant of proportionality. This constant we call **G**, the **Universal Gravitational Constant**.

** G** = 6.67 x 10

^{-11}N m

^{2}kg

^{-2}

**Gravitational force is very weak!** This can be shown by considering two 1 kg masses 1 m apart. The gravitational force between them is given by:

The gravitational force between everyday objects is so small as to be almost irrelevant.

__Variation of ' g' with distance from the Earth's centre __

To understand this work we must recall that '** g**' is the acceleration due to gravity (

**9.8 ms**). This value is for the

^{-2}__of the Earth.__

*surface*

Weight * W* is the force of attraction of the earth on a mass.

For a mass * m*, the weight is given by:

The mathematical treatment depends on **two** assumptions:

1. The value of ** g** is the same at a distance from a mass, whether the mass is in the shape of a spherical shell or concentrated in the centre.

2. The value of ** g** everywhere inside a spherical shell is zero.

**NB** The spherical shell and central mass have uniform density.

First consider a mass * m* on the surface of the Earth. The force of attraction between the mass and the earth is its weight

**This is also equal to the force**

*W.**between the mass and the Earth, given by Newton's Law.*

**F**

(i

where * M_{E}* is the mass of the Earth,

*its radius.*

**r**_{E}

Now let us consider the value of ** g** at a distance

**from the Earth.**

*r*

**case where** ** r** >

**r**_{E}

_{}

If this new value is * g_{r}* , then by similarity with equation (i,

(ii

Dividing equation (ii by equation (i,

(iii

**case where** ** r **<

**r**_{E}

In the diagram the point **X** is **inside** the earth at a distance ** r ** from the centre.

From our initial assumptions, the value of * g_{r}* is a result of the gravity from a sphere of radius

**.**

*r*If * M_{S}* is the mass of the sphere, then by comparison with equation (i ,

(iv

**NB** the effect of matter (in the form of a shell) above point **X** has no effect on the value of **g _{r}**

let us assume that masses have uniform density * ρ* (rho).

Remembering that * m = ρV* , the mass

*of the internal sphere and the mass*

**M**_{S}*of the Earth is given by:*

**M**_{E}

Dividing the first equation by the second,

Substituting for * M_{S}* from equation (iv ,

recalling that

**NB** * g_{r}* =

**when**

*g**=*

**r**

**r**_{E}

__Summary__

So for **inside the Earth**, * g_{r}* is directly proportional to

*. The graph is therefore a straight line through the origin.*

**r**

For **outside the Earth**, * g_{r}* follows a function similar to

**y = x**, where

^{-2}**x**decreases steadily, approaching zero at infinity.

where * r_{E}* and

**are constants**

*g*

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