EXAMPLES 143 



10. Prove that the center of gravity of any portion PQ of a uniform 

 heavy string hanging freely is vertically above the intersection of the 

 tangents at P, Q. 



11. A hemispherical shell has inner and outer radii a, b. Show that 

 the distance of its center of gravity from its geometrical center is 



3 (a + 6)(a 2 + 6 2 ) 

 8 a 2 + ab + b 2 



12. An anchor ring is cut in two equal parts by a plane through 

 its center which passes through its axis. Find the center of gravity of 

 either half. 



13. Prove that the pull exerted by a man in a tug of war is - of his 



b 



weight, where a is the horizontal projection of a line joining his heels to 

 his center of gravity, and b is the height of the rope above the ground. 



14. Prove that a horse weighing W pounds can exert a horizontal pull 

 of Wa/h pounds at a height h above the ground by advancing his center 

 of gravity a distance a in front of its position when he is standing upright 

 on his legs. 



15. A rod of varying density and material is supported by a man's two 

 forefingers, across which it rests in a horizontal position. The man moves 

 his fingers toward one another, keeping them in the same horizontal plane, 

 and allowing the rod to slip over one or both of his fingers. Show that 

 when his fingers touch, the center of gravity of the rod will be between 

 the points of contact of his fingers with the rod. 



16. A semicircular disk rests in a vertical plane with its curved edge on 

 a rough horizontal and an equally rough vertical plane, the coefficient of 

 friction being /*. Show that the greatest angle that the bounding diameter 

 can make with the vertical is 



17. A hemisphere of radius a and weight W is placed with its curved 

 surface on a smooth table, and a string of length l(l<d) is attached to a 

 point on its rim and to a point on the table. Prove that the tension of the 

 string is 3 a _ i 



8 V2 al - I 2 



18. A triangular lamina of weight W is supported by three vertical 

 strings attached to its angular points so that the plane of the triangle is 

 horizontal ; a particle of weight W is placed at the orthocenter of the 

 triangle. Prove that the tensions of the strings are given by 



1 + 3 cot.B cot C ~ 1 + 3 cot C cot A 1 + 3 cot A cotB 2 



