(10) A motion is impressed upon a right circular cone about a given axis through its centre of gravity: to find the position of the invariable plane.

Let the centre of gravity be the origin of co-ordinates, and the axis of the cone be the axis of z: let l, m, n, be the directioncosines of the initial axis of revolution, and let tan a represent the ratio of the diameter of the base to the altitude of the cone. The equation to the invariable plane is

(11) A quiescent circular plate is moveable about its centre of gravity as a fixed point: if a given angular velocity be impressed upon it about a given axis, to find the motion of the plate.

Let a be the given angular velocity, a the inclination of the initial axis of rotation to the plane of the plate. The normal to the plane of the plate will retain a constant inclination to a normal to the invariable plane, and will make a revolution in space in a time equal to

2π

w (1+3 sin2 a) *

Griffin Ib. p. 41.

(12) A solid of revolution is revolving initially about an instantaneous axis, which passes through its centre of gravity, and is inclined at an angle y to its axis of figure, which is initially inclined at an angle a to an axis fixed in space: given that A cot a = C cot y, where C, A, are the moments of inertia about the axis of figure and a perpendicular axis through the centre of gravity, to determine the motion of the axis of figure. The axis of figure will describe a circular right cone about the fixed axis. Griffin: Ib. p. 41.

(13) A solid of revolution, moveable about its centre of gravity, is originally put in motion about an axis the moment

of inertia about which is given: to determine the nature of the subsequent motion.

Let be the given moment of inertia, C the moment of inertia about the axis of figure, and A that about an axis, perpendicular to the axis of figure, through the centre of gravity. Then

(1) The instantaneous axis will describe in a space a circular cone, the vertical angle of which is greatest when Q is a harmonic mean between A and C.

(2) The axis of figure will describe in space a circular cone the vertical angle of which is equal to

(3) The instantaneous axis will describe a circular cone, relatively to the axis of figure, the vertical angle of which is equal to

(14) A right circular cone is moving about its centre of gravity as a fixed point: supposing the initial inclination of the instantaneous axis to the axis of figure to be known, to find the path of the vertex of the cone.

Let a be the initial inclination of the instantaneous axis to the axis of figure, ẞ the semi-angle of the cone, and h its altitude. The vertex of the cone will describe in space a circle the radius of which is equal to

3
h tan a (4 + cot2 ß).
32

(15) A body is moving about a fixed point, two of the principal moments of inertia at the fixed point being equal: supposing the body to be acted on only by a couple, the moment of which is an explicit function of the time, about the

axis of unequal moment, to find the angular velocities about the axes at any time.

Let a be the moment of inertia about one of the axes, b that about each of the other two: let w1, w1, w,, be the angular velocities at any time t, a,, a, a,, being the initial values of these velocities: let T denote the moment of the couple. Then, a2 denoting a2 +α,2,

(16) A rigid lamina, in the form of a loop of a lemniscate, not acted on by any force, is started with a given angular velocity about one of the tangent lines at its nodal point, the nodal point being fixed: to find the ratio of its greatest to its least angular velocity.

The required ratio is equal to

4

1+

3п

SECT. 3. Several Bodies.

(1) To a wheel and axle are attached weights P and Q, (fig. 200), which are not in equilibrium: to determine their motion and the tensions of the strings by which the weights are suspended.

=

=

Through C, the centre of the wheel and axle, draw the horizontal line ACB meeting the strings in A and B; let AC=a, BC= a'; m = the mass of P, m' that of Q, μ that of the wheel and axle together; k = the radius of gyration of the wheel and axle about their common axis; AP= x, BQ = x', T= the tension of AP, T" the tension of BQ; 0 = the angle

=

through which the wheel and axle have revolved at the end of the time t about their common axis.

Then, for the motion of P, we have

and, for the rotation of the wheel and axle,

Substituting the values of Tand 7" from (4) and (5) in the equation (3), we get

whence is immediately obtained in terms of t, the initial

do
dt

values of and being supposed to be known.

From (4) and (6) we have

(2) Two equal uniform rods AC, BC, (fig. 201), having a compass joint at C, are laid in a straight line upon a horizontal plane. A string CDP, to one end of which is attached a weight P greater than that of either rod, passes over a smooth pin D above the plane, and is fastened at its other end to C, which is vertically beneath the pin: to determine the motion of P.

Let AC=2a= BC, ▲ CAB=0; R = the reaction of the plane at each of the points A and B; T= the tension of the string; Sthe mutual action of the two rods at the joint, which will evidently take place in a horizontal line parallel to AB; m = the mass of each of the rods, μ the mass of the weight P. Let G be the centre of gravity of the rod AC; draw GH, CE, at right angles to AB; let EH=x, GH=y, k = the radius of gyration of AC about G.

=

Then, for the motion of the rod AC, we have, resolving forces horizontally,

dx
dt

m = S......

(1);

resolving vertically, T being the force exerted by the string on

Also, for the motion of P, the increment of DP being double that of GH,

Multiplying the equation (2) by a cos 0, we have

·(4).

and therefore, adding this equation to the equation (3),