For the third part of the question, in the variationcompass and the dip-needle, the planes are at right angles, dW dW are the couples interfering with the effect of

and

de аф

the earth's directive force.

1. DEMONSTRATE formulae involving polar coordinates for the position of the centre of inertia of a plane lamina and of a solid.

If the density at any point of a circular disc whose radius is a vary directly as the distance from the centre, and a circle described on a radius as diameter be cut out, prove that the centre of inertia of the remainder will be at a 6a from the centre.

distance

15π 10

Therefore, if C is the centre of the large circle, and B the centre of inertia of the circle cut away,

2. Find expressions for the accelerations of a moving point estimated (a) along and perpendicular to the radius vector (B) along the tangent and normal.

If a curve be described under a force P tending to the pole and a normal force N, prove that

dr

2

2

d

p' d ( Nr dc) + y2 & ( Pp dr) + 2Py2 = 0,

dr

whence the result.

dr

3. A particle moves under the action of given forces on a given smooth surface, shew how to determine the motion and the pressure on the surface.

Given the resultant impressed force and the velocity of the particle at any point, determine by a geometrical construction the osculating plane and the centre of curvature of the path on the surface.

Let S be the surface (fig. 54), M the particle of mass m projected in the direction MT with velocity v, and let F be the resultant impressed force represented by MF.

If N be the normal reaction of the surface represented by MN, we can consider the particle as moving freely under the action of the resultant K, represented by MK, of F and N; KMT will then be the osculating plane of the path.

N is unknown, but is determined from the condition that

mv2

if MI be taken equal to where R is the radius of 2. Ꭱ

curvature of the normal section made through MT; then MI is equal to the algebraical sum of MN and the projection of MF on MN.

The osculating plane being then determined, if a line be drawn through I at right angles to the plane IMT, it will, by Meunier's theorem, meet the osculating plane in H, the centre of curvature of the path.

iv. Form the equation of motion of a rigid plate of any form consequent upon one of its points being constrained to move in a given manner in the plane of the plate. Integrate it for the special cases of uniform rectilinear, and uniform circular, motion of the point.

Apply your results to explain the action of a flail, gravity being neglected.

Let, be the coordinates of the point given in terms of t; xa cose, y=a sine the relative coordinates of the centre of inertia of the plate; X, Y the component forces applied by the constraint. Then

M (§ + x) = X, M (+ÿ) = Y, and MÖ = Xy – Yx. Therefore 0=(+ x) y − (ï + ÿ) x = §y — ïx — a2ë,

(k2 + a2) Ö = ¿y − ŋx.

(1) In the case of uniform rectilinear motion of the point =0, 70; therefore = 0, and the plate rotates uniformly. (2) In the case of uniform circular motion of the point

=-bn cos nt, n=-bn2 sinnt;

These two cases explain the flail. The handle is worked. so that the joint has the circular motion until the striker gains its maximum angular velocity; the motion of the joint is then changed into the rectilinear motion, when the striker continues to move with the same angular velocity until the blow is delivered.

v. Write down Euler's equations which give the angular velocities of a rigid body about its principal axes, and interpret the various terms.

If a constant couple be applied about the axis of symmetry of a body supported at its centre of inertia, and initially rotating about an axis perpendicular to that of symmetry, determine the motion completely; and shew that the cone described in the body by the instantaneous axis has the equation

A-C Q°C

where N is the couple, the initial angular velocity.

The equations of motion are.

hence the equation of the cone described by the instantaneous

axis.