126

STATICS.

Eliminating w from this by (4) we have

cos 9 sin 2 (a + 9) = - sin 2 a,x a

from which 9, and therefore the position of the beam, is to bedetermined.

If a = 90°,

a

Pkob. 4. A sphere and cone in contact rest, as in fig. 68,on two inclined planes, the intersection of which is a horizontalline : required the angle of the cone and the position of equi-librium.

W, W' the weights of the sphere and cone : R the reactionat B : P the mutual action at E : the resultant of the reactionsof the plane on the base of the cone must act at some point D,let Q be this resultant: CD = x : G the centre of gravity ofthe cone : rad. of sphere = a, Ge = x, e being the point wherethe normal at E cuts the axis of the cone: 2 9 = the angle ofthe cone : a, fi the angles the planes make with the horizon.

For the sphere, IE — JR cos /3 4- P sin (a — 0) = 0 . (l),

. R sin (3 - P cos (a - 9) - 0 . (2).

The equation of moments is an identical equation.

For the cone, W'— Q cos a — P sin (a — 9) = 0.( 3 ),

Q sin a - P cos (a - 9) = 0.( 4 ),

moments about G, Qx — P% cos 9 = 0 .( 5 ).

These five equations involve six unknown quantities: if therebe a sixth equation it must be a relation connecting the geome-trical quantities involved in these five equations: but a littleconsideration will shew us that no necessary connexion existsbetween any two of oc, 9, % : hence the problem is indeterminate.By examining the equations we perceive that the first fourinvolve only the four unknown quantities P, R, Q, 9 : hence