Thus we have determined the distance from Ox of the centre of gravity of four heavy particles; and in the same manner we may proceed whatever be the number of heavy particles.

146. In the same way if the distances of A, B, C, and D from a second straight line, as Oy, in the same plane be given, we can deduce the distance of the centre of gravity of the system from the same straight line.

And when we know the distance of the centre of gravity from two straight lines in the plane we can determine the position of the centre of gravity; for it will be at the point of intersection of straight lines parallel to Ox and Oy and at the respective distances from them which have been found.

It is easy to extend our investigation to the case in which the heavy particles are not all in one plane; see Art. 119.

EXAMPLES. IX.

1. If two triangles are on the same base, shew that the straight line which joins their centres of gravity is parallel to the straight line which joins their vertices.

2. A rod 3 feet long and weighing 4 lbs. has a weight of 2 lbs placed at one end; find the centre of gravity of the system.

3. A quarter of a triangle is cut off by a straight line drawn parallel to one of the sides: find the centre of gravity of the remaining piece.

4. Find the centre of gravity of a uniform circular disc out of which another circular disc has been cut, the latter being described on a radius of the former as diameter.

5. If three men support a heavy triangular board at its three corners, compare the force exerted by each man.

6. Shew that the centre of gravity of a wire bent into a triangular shape coincides with the centre of the circle inscribed in the triangle formed by joining the middle points of the sides of the original triangle.

7. If the centre of gravity of a triangle be equidistant from two angular points of the triangle, the triangle must be isosceles.

8. If a straight line drawn from an angular point through the centre of gravity of a triangle be perpendicular to the opposite side, the triangle must be isosceles.

9. A triangle ABC has the sides AB and BC equal; a portion APC is removed such that AP and PC are equal compare the distances of B and P from AC in order that the centre of gravity of the remainder may be at P.

10. A heavy bar 14 feet long is bent into a right angle so that the lengths of the portions which meet at the angle are 8 feet and 6 feet respectively: shew that the distance of the centre of gravity of the bar so bent from the point which was the centre of gravity when the bar was straight, 9/2 feet.

is

7

11. If the centre of gravity of three heavy particles placed at the angular points of a triangle coincides with the centre of gravity of the triangle, the particles must be of equal weight.

12. Two equal uniform chains are suspended from the extremities of a straight rod without weight, which can turn about its middle point: find the position of the centre of gravity of the system, and shew that it is independent of the inclination of the rod to the horizon.

13. The middle points of two adjacent sides of a square are joined and the triangle formed by this straight line and the edges is cut off: find the centre of gravity of the remainder of the square.

14. If n equal weights are to be suspended from a horizontal straight line, and a given length l of string is to be used, determine the distance of the centre of gravity of the weights from the straight line.

15. If the sides of a triangle be 3, 4, and 5 feet, find the distance of the centre of gravity from each side.

16. A piece of uniform wire is bent into the shape of an isosceles triangle; each of the equal sides is 5 feet long, and the other side is 8 inches long: find the centre of gravity.

17. Find the centre of gravity of the figure formed by an equilateral triangle and a square, the base of the triangle coinciding with one of the sides of the square.

18. Two straight rods without weight each four feet long, are loaded with weights 1lb., 3 lbs., 5lbs., 7lbs., 9lbs. placed in order a foot apart: shew how to place one of the rods across the other, so that both may balance about a fulcrum at the middle point of the other.

19. A rod of uniform thickness is made up of equal lengths of three substances, the densities of which taken in order are in the proportion of 1, 2, and 3: find the position of the centre of gravity of the rod.

20. A table whose top is in the form of a right-angled isosceles triangle, the equal sides of which are three feet in length, is supported by three vertical legs placed at the corners; a weight of 20 lbs. is placed on the table at a point distant fifteen inches from each of the equal sides: find the resultant pressure on each leg.

X. Properties of the Centre of Gravity.

147. When a body is suspended from a point round which it can move freely it will not rest unless its centre of gravity be in the vertical line passing through the · point of suspension.

For the body is acted on by two forces, namely its own weight in a vertical direction through the centre of gravity, and the force arising from the fixed point. The body will not rest unless these two forces are equal and opposite. Therefore the centre of gravity must be in the vertical line which passes through the point of suspension.

148. The preceding Article suggests an experimental method of determining the centre of gravity of a body which may sometimes be employed. Let a body be suspended from a point about which it can turn freely, and let the direction of the vertical line through the point of suspension be determined, Again, let the body be suspended from another point so as to hang in a different position, and let the direction of the vertical line through the point of suspension be determined. The centre of gravity is in each of the two determined straight lines, and is therefore at their point of intersection.

149. When a body can turn freely round an axis which is not vertical, it will not rest unless the centre of gravity be in the vertical plane passing through the axis.

The weight of the body may be supposed to act at the centre of gravity. Resolve it into two components at right angles to each other, one component being parallel to the axis. The component parallel to the axis will not produce nor prevent motion round the axis; but the other component will set the body in motion round the axis, unless the centre of gravity be in the vertical plane passing through the axis.

150. A body which is suspended from a fixed point by means of a string will not rest unless its centre of gravity be below the fixed point to which the string is fastened.

But a body which can turn freely round a fixed point rigidly connected with it may rest with its centre of gravity either vertically above or vertically below the fixed point. And in like manner when a body can turn freely round a fixed axis which is not vertical it may rest with its centre of gravity either above or below the axis. There is an important difference between the two positions of equilibrium, which is shewn by the following proposition.

151. When a body which can turn freely round a fixed point is in equilibrium, if it be slightly displaced it will tend to return to its position of equilibrium or to recede from it according as the centre of gravity is above or below the fixed point.

This may be taken as an experimental fact; or it may be established thus:

Let O be the fixed point, G the centre of gravity of the body. Draw GH vertically downwards. The weight of

K

G

K

H

the body acts along GH; resolve it into two components at right angles to each other, one along the straight line which joins the centre of gravity with the fixed point: let GK be the direction of the other component.

When G is nearly above O the former component acts along GO; and thus the latter obviously tends to move the body away from the position in which G is vertically above 0.

When G is nearly below O the former component acts along OG; and thus the latter obviously tends to move the body towards the position in which G is vertically below 0.