142 CENTER OF GRAVITY 



GENERAL EXAMPLES 



1. A plane quadrilateral A BCD is bisected by the diagonal AC, and 

 this diagonal is divided in the ratio a : b by the diagonal BD. Prove 

 that the center of gravity of the quadrilateral lies in AC and divides it 

 into two parts in the ratio 2 a + & : 2b + a. 



2. A uniform wire is bent into the form of a circular arc and the two 

 bounding radii, and the center of gravity of the whole is found to be 

 at the center. Show that the angle subtended by the arc at the center is 

 tan-i(-f). 



3. The three feet of a circular table are vertically below the rim and 

 form an equilateral triangle. Prove that a weight less than that of the 

 complete table cannot upset it. 



4. A triangular table is supported by three legs at the middle points 

 of its sides, and a weight W is placed on it in any position. It is found 

 that the table will just be upset if a weight P is placed at one angular 

 corner. The corresponding weights needed to upset it at the other corners 

 are Q, E. Prove that P + Q + R is independent of the position of the 

 weight W. 



5. Weights are nailed to the three corners of a triangular lamina, each 

 proportional to the length of the opposite side of the triangle, and of com- 

 bined weight equal to the original weight of the lamina. Show that the 

 center of gravity of the triangle is at the center of the nine-point circle. 



6. A uniform triangular lamina of weight W and sides a, b, c is sus- 

 pended from a fixed point by strings of lengths Z 1? / 2 , l s attached to its 

 angular points. Show that the tensions of the strings are 



WU lt WM Z , Wkl s , 

 where k = [3 (tf + ij + 1$) - a 2 - & 2 - c]~*. 



7. Explain how a clock hand on a smooth pivot can be made to show 

 the time by means of watchwork, carrying a weight round, concealed in 

 the clock hand. 



8. A spindle-shaped solid of uniform material is bounded by two right 

 circular cones of altitudes 6 and 2 inches with a common circular base of 

 radius 1 inch. It is suspended by a string attached to a point on the rim 

 of the circular base. Find the inclination of the axis of the spindle to the 

 vertical when it is hanging freely. 



9. A pack of cards is laid on a table, and each projects beyond the one 

 below it in the direction of the length of the pack to such a distance that 

 each card is on the point of tumbling, independently of those below it. 

 Prove that the distances between the extremities of successive cards will 

 form a harmonic progression. 



