284. A cube, the edges of which are twelve equal uniform rods hinged together, is hung up by one corner, the cube form being maintained by a string joining this corner with the lowest corner. It is required to find the initial change of stress at the point of support when the string is cut.

K

L

A

B

The corner O being the point of support and the diagonal OD vertical, it is clear that the initial angular accelerations of OA, OB, OC will be respectively in the planes AOD, BOD, COD and will be equal to each other; and further that the angular accelerations of all the other rods will be the same and will be, respectively, in parallel planes.

If a represent this initial angular acceleration, 2aw, which we shall call 2f, will be the linear acceleration of A in the direction AD, of B in direction BD, and of C in direction CD.

Taking accelerations parallel to OA, OB, and OC, we obtain the following forms, where K and L are the centres of the rods CE, ED.

Now suppose a displacement made by slightly increasing the length of OD, so as to turn every rod through a small angle 0.

The displacements of the various points follow the law of the accelerations and are of the same forms, replacing o by 0.

Observing that there are six rods under the same conditions as CE, three under the same conditions as ED, and three other rods OA, OB, OC, the equation of virtual work is

6mfa0 (2 + 1⁄2 + 1) + 3mfa0 (1⁄2 +8 + 2)

for the displacement of D is the resultant of the displacements of A, B, and C.

From this equation we obtain

25aw = 3g√6,

and since the acceleration of D is 2aw/6, it follows that the acceleration of G is aw/6 and is therefore 18g/25.

Hence it follows that the diminution of stress at O is 18/25 of the total weight of the system.

The initial stresses at the several joints can be obtained by giving independent displacements to the several rods, breaking the connections at the different joints.

EXAMPLES.

1. A number of concentric rough spherical shells, fitting each other, so as to have sliding contact, are set rotating about different axes; find the ultimate angular velocity of the system when their relative motions are destroyed by friction.

2. A sphere is projected horizontally on an inclined plane, the surface of which is perfectly rough; shew that its centre will describe a parabola.

3. Two particles of masses m, 2m are fixed to the ends of a weightless rod of length 2a which is freely moveable about its middle point. Prove that if @ be the inclination of the rod to the vertical when the particles are moving with uniform angular velocity w, 3w'a cos 0 = g.

4. A solid rectangular parallelepiped with edges of length a, b, c, is acted on by instantaneous couples with axes parallel to these edges and of moments proportional to pqr; shew that the direction cosines of the instantaneous axis of rotation are in the ratio

5. A rod, of mass 3m and length 2a, is moveable in a vertical plane about its middle point, and carries at one end a particle of mass m; if the vertical plane be made to revolve, with uniform angular velocity w, about the vertical through the middle point, prove that the equation of motion of the rod is

2a0-2aw2 sin 0 cos 0+g sin 0 = 0.

6. A rigid body moveable about a fixed point is struck by a blow of given magnitude at a given point: if the angular velocity thus impressed upon the body be the greatest possible, prove that, a, b, c, being the coordinates of the given point in relation to the principal axes through

the fixed point, and l, m, n, being the direction-cosines of the blow,

A, B, C, being the moments of inertia of the body about the principal axes at the fixed point.

7. If an octant of an ellipsoid bounded by three principal planes be rotating about the axis a with angular velocity w, and if this axis suddenly become free, and the axis b fixed, shew that the new angular velocity is 2abw/π (α3 + c3).

8. A rectangular parallelopiped is dropped on to a smooth floor so that one angular point first comes in contact; if the edges be 2a, 2b, 2c and equally inclined to the vertical at the instant of striking, find the impulse sustained by the floor.

9. A ring rests upon two smooth horizontal bars which in the position of equilibrium subtend an angle 2a at the centre; shew that, if the ring be disturbed by twisting it through a small angle about its vertical diameter, the length of the simple isochronous pendulum will be c cot a cosec a.

10. A heavy, uniform, and inextensible string is in equilibrium in the form of a horizontal ring on a smooth sphere; prove that, if it be cut at a point A, the initial change of tension at a point P will be to the weight of the string in the ratio

cos h ( cos a): 2π cot a cos h (π cos α),

a being the angular distance of the string from the vertex of the sphere, and π- the angle subtended at the centre of the ring by the arc PA.

11. A frame consists of four equal uniform rods looselyjointed at their ends so as to form a square, and one of the rods carries a light ring fastened to it at its middle point. The frame moves with uniform velocity on a table. All kinds of friction being neglected, prove that when a vertical bolt is shot through the ring the frame will be brought absolutely to rest.

B. D.

25

12. A square lamina is revolving about a vertical diagonal, the highest point of which is fixed, with the angular velocity w. If suddenly one of the angular points in motion becomes fixed, prove that the square will just revolve round the fixed side, if aw2 = 96g√2, where a is the length of a side of the square. Prove also that the impulses at the fixed points are in the ratio of 3 to 5.

13. One end of a heavy rod rests on a horizontal plane and against the foot of a vertical wall, the other end rests against a parallel vertical wall, all the surfaces being smooth. Shew that if it slips down, the angle through which it turns round the common normal to the vertical walls is given by the equation

$2 (1 + 3 cos2) = C – 6g sin ¢/√a2 — b2,

where 2a is the length of the rod, and 26 the distance between the walls.

14. A smooth plate inclined at an angle & to the horizon is made to rotate about a vertical axis AB with uniform angular velocity w. A rod of mass m is compelled by guides to be always vertical, and at a distance r from AB, while it rests with one end in contact with the plate, sliding up and down as the latter rotates. Shew that, if the rod be initially in its lowest position, the pressure on the plate at the end of the time t will be

m (g cos &+rw2 cos wt sin p) sec2p.

15. A rhombus of mass M, formed of four equal rods jointed together, is moving in the direction of a diagonal with velocity u, and suddenly a particle of mass m becomes affixed to one end of the diagonal; prove that, if 2a be the length of each rod, the angular velocity a suddenly acquired by each rod is such that

2aw {M+m (1+3 sin2 a)} = 3mu sin a,

and that the kinetic energy lost is

Mmu2/{M+m (1 + 3 sin2a)}.