EXAMPLES. 325 



93. A uniform equilateral triangular board is suspended by three equal 

 cords from the corners of a similar triangle in a horizontal plane ; prove that, 

 if one of the cords is cut, the tensions in the remaining two are instantly 

 diminished in the ratio 3 sin 2 a : 2 -f 4 sin 2 a, where a is the angle which the 

 plane through the two cords makes with the horizontal. 



94. A thin uniform rectangular board hinged along a line in itself 

 parallel to one side is opened out to any angle and placed on a smooth 

 horizontal plane so that the cross section of the board made by a vertical 

 plane perpendicular to the hinge is a triangle ABC, of which the side AB is 

 in the horizontal plane, and the angle C in the hinge. Prove that C starts to 

 move in a direction making with the vertical an angle whose tangent is 

 tan A tan B tan ^ (A - B), and find its initial vertical and horizontal 

 accelerations. 



95. A circular ring hangs in a vertical plane on two pegs in a horizontal 

 line subtending an angle 2a at the centre. One peg is suddenly removed. 

 Find the pressure on the remaining peg when it is (1) smooth, (2) rough, and 

 prove that these pressures are in the ratio 1:^(1 + jtan 2 a). 



96. Four uniform rods two of length 2a and two of length 26 whose 

 masses are proportional to their lengths are freely jointed so as to form a 

 parallelogram. One of the rods of length 2a is free to turn about a pivot at 

 a distance c from its middle point, and is initially held in a horizontal 

 position so that the figure is a rectangle, and is let go. Prove that the 

 initial angular acceleration of each of the horizontal rods is 



97. A sphere resting on a rough horizontal plane is divided into an in- 

 definitely great number of solid segments by planes through a common 

 diameter and is kept in shape with this diameter vertical by means of a band 

 round the horizontal great circle. Prove that, if the band is cut the pressure 

 on the plane is instantly diminished by the fraction 457r 2 /2048 of itself. 



98. Two uniform rods AB, BC of masses m, m' and lengths 2a, 26 are 

 freely jointed at B. The end A is fixed and the system is supported by a 

 string attached to C in a position in which AB, BC make angles a and /3 with 

 the vertical. Prove that, if the string is cut, the initial angular accelerations 

 of AB and BC are 



| (m -f 2w') sin a - 2m' sin /3 cos (a ) g 

 ~ IJL ( m + 3^') _ 4 m ' C os 2 (a - /3) a' 



, $ (m + 3m') sin 3 - 2 (m + 2m') sin a cos (a - ft) g 



L& ( m + 3m') - 4m' cos 2 (a - 0) ~ 6* 



99. A uniform rod of length 2a and weight W rests on a rough horizontal 

 plane with its pressure on the plane uniformly distributed. A horizontal 

 force P, large enough to produce motion, is suddenly applied at one end 

 perpendicularly to the length of the rod. Prove that the rod begins to turn 



