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This result shows that the shell attracts the particle at C in the same manner as if the mass of the shell were condensed into its centre.

3. It follows also that a sphere, which is either homogeneous or consists of concentric spherical shells of uniform density, will attract the particle C in the same manner as if the whole mass were collected at its centre.

PROP. To find the attraction of a homogeneous spherical shell of small thickness on a particle situated within it.

4. We must proceed as, in the last Proposition; but the limits of y are in this caser -c and r+c; hence,

πρrdr

72 attraction of shell

1

dy ca

ya Trprdr

(20— 2c) = 0;

therefore the particle within the shell is equally attracted in every direction.

5. This result may easily be arrived at geometrically in the following manner. Through the attracted point suppose an elementary double cone to be drawn, cutting the shell in two places. The inclinations of the elementary portions of the shell, thus cut out, to the axis of the cone will be the same, the thickness the same, but the other two dimensions of the elements will each vary as the distance from the attracted point; and therefore the masses of the two opposite elements of the shell will vary directly as the square of the distance from that point, and consequently their attractions will be exactly equal, and being in opposite directions will not affect the point. The whole shell may be thus divided into pairs of equal attracting elements and in opposite directions, and therefore the whole shell has no effect in drawing the point in any one direction more than in another.

6. The results of these two Propositions are so simple and beautiful, that it is interesting to enquire whether these

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The attraction of the whole shell evidently acts in CO.

Let OP revolve about through a small angle do in the plane MOP; then rdo is the space described by P. Again, Iet OPM revolve about OC through a small angle do, then r sin 0 do is the space described by P. And the thickness of the shell is dr. Hence the volume of the elementary portion of the shell thus formed at P equals rdo .r sin . dr ultimately, since its sides are ultimately at right angles to each other.

Then, if the unit of attraction be so chosen, that it equals the attraction of the unit of mass at the unit of distance, the attraction of the elementary mass at P on C in the direction CP pré sin drdodo

,p the density of the shell ;

ya .. attraction of P on C in CO prø sin 0 drdodo c-r cos e

We shall eliminate 0 from this equation by means of

y' = c + gol – 2cr cos 6,
d
y

y? 22
C-r cos O =
dy

2c

c2 - 22

(1+ :) dy do. To obtain the attraction of all the particles of the shell we integrate this with respect to $ and y, the limits of $ being 0 and 27, those of y being c-p and c+r;

ca .:. attraction of shell on C

1+ 2c

dy do ca

Trprdr

dy (2r + 2r) c

ya

ca 4m prédr mass of shell

ca

.. sin o

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.. attraction of P on C in CO = predlo a

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This result shows that the shell attracts the particle at C in the same manner as if the mass of the shell were condensed into its centre.

3. It follows also that a sphere, which is either homogeneous or consists of concentric spherical shells of uniform density, will attract the particle C in the same manner as if the whole mass were collected at its centre.

PROP. To find the attraction of a homogeneous spherical shell of small thickness on a particle situated within it.

4. We must proceed as in the last Proposition; but the limits of y are in this case r c and r+c; hence,

#prder

ric

attraction of shell =

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Trprdr

(20 – 2c) = 0;

ca therefore the particle within the shell is equally attracted in every direction.

5. This result may easily be arrived at geometrically in the following manner. Through the attracted point suppose an elementary double cone to be drawn, cutting the shell in two places. The inclinations of the elementary portions of the shell, thus cut out, to the axis of the cone will be the same, the thickness the same, but the other two dimensions of the elements will each vary as the distance from the attracted point; and therefore the masses of the two opposite elements of the shell will vary directly as the square of the distance from that point, and consequently their attractions will be exactly equal, and being in opposite directions will not affect the point. The whole shell may be thus divided into pairs of equal attracting elements and in opposite directions, and therefore the whole shell has no effect in drawing the point in any one direction more than in another.

6. The results of these two Propositions are so simple and beautiful, that it is interesting to enquire whether these

2

A"

TATIONS.

The attraction of t*
Let OP revoly

er or not to the law of the inverse

determine this is the object of the plane MOP; tl let OPM revol r sin do is t]

stiraction of a homogeneous spherical the shell is di of the shell

parut it; the law of attraction being repremately, sinc other.

u is exactly analogous to that given Then, if

v to alter the law of attraction. Then the attrac the attra direction

the distance.

• Co

'+0=)$(9) dy (integrated by parts) ,1+0=1956 (9) dy 2{4$() dy} dy]

.. att

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gjere) $(3) 24 (9) +const.} suppose, fet16; (+r) -54 (c+r) -- * 0,60-v) + x =r}} d lys (c+r) — (c— r)

(e –r}, táis latter form being introduced merely as an analytical artifice to simplify the expression.

PROP. To find the attraction of the shell on an internal particle, with the same law.

aprar

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C

8. The calculation is the same as in the last Article, except that the limits of y are r- c and r+c:

... attraction = 2#prdr (mitte (+c) – 3 x (+ +c)

+2002 (ro) + (r-cy} – 2xpride o [ro

PROP. To find what laws of attraction allow us to suppose a spherical shell condensed into its centre when attracting an external point.

9. Let (r) be the law of force; then if c be the distance of the centre of the shell from the attracted point and r the radius of the shell, and

(n) = S {r 5$ (r) dr} dr, then the attraction of the shell

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But if the shell be condensed into its centre, the attraction

: 4mpra dr $ (c);

24 (c+r) - - (c () :

dc

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difc

d'fc But :c5$ (c) dc, 5$ (c) dc + co (c), dc

dca
dync

doc
dcs
= 26c+c

do:
therefore by the first of the above equations of condition

2 doc
фс + const. = 3A,

de

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