and the equilibrium is stable or unstable according as his

Obs. If G', M coincide the displacement being very

(ii) If the surface of BAC be plane-as in the case of a solid resting with its plane base upon a curved surface-r∞, and the equilibrium is stable or unstable according as

h< or > R.

(iii) If the surface of QAR be plane-as in the case of a solid resting

B

R

with its curved surface upon a horizontal plane-R = ∞, and the equilibrium will be stable

or unstable according as

85. The following is an example of finding the centre of gravity which leads to some useful results.

To find the centre of gravity of n equal particles arranged at equal intervals along a circular arc.

Let O be the centre of the circular arc AB, along which the n equal particles A, P, Q,

R,... B are arranged;

so that (n-1) = 2x......(i).

Then if (x, y) (X2, Y2).......

be the co-ordinates of the suc

cessive particles A, P, Q... re

ferred to Ox, Oy as rectangular axes, we have (Art. 74)

x=

=

α

n

= = {1 + cos + cos 20 + ... + cos (n − 1) 0}

n

1

sin 0

212

(by Trigonometry)

by substituting for in terms of a from (i).

(ii)

i. e. G lies in the line OG which bisects the AOB, and (ii), gives its distance from 0.

86. COR. From the preceding investigation we may deduce some useful results.

1

If the number of particles n be supposed to become in

I. Since a uniform material circular arc may be regarded

as a series of equal particles at small equal intervals,-if AB be a uniform circular arc of which O is the centre, and G the centre of gravity, 2a the circular measure of the ▲AOB and A0= a;

then we infer from the above that OG bisects the AOB,

B

G

up

of a

B

II. Again, since we may regard the circular arc AB as the limit of a polygon of a very large number of sides, we may regard the circular sector AOB as made very large number of triangles having a common vertex at 0, and the sides of this polygon for their bases,—and if Or be the distance from O of the centre of gravity of any one of these triangles Opq, we shall have (when the pOq is taken very small)

Or Opa, in the limit,

g

A

and the centre of gravity g of the sector AOB will coincide with the centre of gravity of a uniform circular arc ab whose

2

radius =.a.

i. e. Og bisects the AOB, and Og

the sector becomes a semicircle, and in this case

III. The centre of gravity G of the sector AOB being known, as well as G, that of the triangle AOB,-we can easily (Art. 71) find G, the centre of gravity of the circular segment 0 ABC.

For AAOB = a2 sin a cos a,

sector AOB = a2a,

segment ABC a2 (a

B

G G G

= ·

sin à cos a).