6. Shew that it is necessary and sufficient for the equilibrium of a system of forces in one plane on a rigid body that the sums of the resolved parts of the forces in two directions and the sum of the moments of the forces about one point should separately vanish.

Four forces are applied at the angles of a rigid rectangle ABCD and in its plane, so as to balance. If X1, X2, X3, X4 are the components in the direction AB of the forces at A, B, C, D respectively and Y, Y2, Y, Y, the components in the direction BC, shew that

BC(X1 + X2) = − BC(X3 + X4)

= AB(Y1 + Y) = — AB(Y2 + Y3).

7. Explain in what cases the reactions on a bar at the two joints which connect it with a frame can be taken as acting along the line joining the joints.

In the second part of question 6, suppose that the rectangle is formed of five bars AB, BC, CD, DA, AC smoothly jointed at the corners. Shew that the stress in the diagonal bar AC is

(X + X1)AB|AC+ (Y2 + Y)BC AC.

8. Shew that the centre of mass of a homogeneous solid tetrahedron is that of four equal particles at its corners.

Shew that the centre of mass of a homogeneous solid pyramid whose base is a parallelogram is that of four equal particles at the angles of the base, together with a particle of four-thirds the

mass of each at the vertex.

9. Shew that the resultant vertical pressure on a surface immersed in heavy liquid is equal to the weight of the superincumbent liquid, and acts through the centre of mass of that liquid.

A hollow sphere is filled with liquid through a small hole at the highest point. Shew that the resultant pressure on the upper half of the sphere is one-quarter of the weight of liquid in the sphere. What is the resultant pressure on the whole sphere?

10. Find the centre of pressure of a triangle immersed in heavy liquid with one side in the surface (a) when the atmospheric pressure is neglected, (b) when it is included.

11. An open cylindrical vessel of height 1 foot and cross-section 1 square foot is inverted and forced down into water until its bottom is in the surface of the water. Shew that if the height of the water barometer is 33 feet the water rises 343 inches in the vessel, and find in lbs. weight the force required to hold the vessel in this position.

MIXED MATHEMATICS.-PART III.

The Board of Examiners.

1. Give an analytical discussion of the principle of Virtual Work.

A lamina, constrained to remain in a fixed plane, is acted on in that plane by forces which maintain their magnitudes and directions in space

when the lamina is displaced. Find the work of the forces for any displacement of the lamina, and deduce (a) the conditions of equilibrium, (b) the condition of rotational stability.

2. Investigate the general conditions of equilibrium of a body under any system of forces, using rectangular coordinates.

The sole constraint of a body is that a plane Xx + μy + vz = 0 fixed in the body passes through a fixed line x1 = y/m = z/n in space. Write down the analytical conditions of equilibrium under a system of forces such as (X, Y, Z) at (x, y, z).

3. Investigate the equation y = c cosh a/c of the common catenary.

If the catenary is made stiff, and is jointed at the lowest point and at the ends of its span, find the stresses in it due to an additional load which is uniformly distributed with respect to the (horizontal) span.

4. Investigate formulæ for the acceleration of a particle in plane polar coordinates.

A particle P is constrained to move in a circle around a fixed point O1. Find formulæ for the acceleration of P along and perpendicular to the radius OP (r) from another fixed point O, in terms of the single dependent variable r.

5. Investigate the rectilinear motion of a particle due to a central force varying as the distance and a resistance as the velocity.

Find the effect of a simple harmonic oscillation of the centre of force on the motion of the particle.

6. Discuss the finite oscillations of a simple unresisted pendulum.

The bob of the pendulum is connected with a fixed point vertically below the point of suspension by an elastic string which is of its natural length when the bob is in its lowest position. Discuss the oscillations in a vertical plane.

7. Investigate the general equations of motion of a rigid body in two dimensions.

A homogeneous solid cylinder of radius a is set spinning about its axis with angular velocity w, and is then placed with its axis horizontal on a rough inclined plane of elevation a. Examine the subsequent motion for the case in which the line of contact of the cylinder is initially moving up the plane.

8. Prove Lagrange's general equations of motion for a system of any number of degrees of freedom.

A uniform rod is suspended from a fixed point by an inextensible string attached to its middle point. Discuss the possible motions, the system being supposed to remain in a vertical plane.

MIXED MATHEMATICS AND NATURAL

PHILOSOPHY.

SCHOOL OF ENGINEERING.-SECOND YEAR.

The Board of Examiners.

PASS AND FIRST HONOUR PAPER.

Civil Engineering Candidates take sections A and B. Mining Engineering Candidates take section B.

A.

1. Investigate the moment of inertia of a homogeneous solid rectangular block about a line through its centre of mass and perpendicular to one pair of faces.

A lamina has the form of a rectangle of breadth 2a and length 27 with added semicircular ends. Find its moment of inertia about a line through its centre of mass and perpendicular to its plane.

2. Prove that the kinetic energy of a system is the sum of that due to the whole mass moving with the centre of mass and that due to the motion relative to the centre of mass.

A smoothly jointed rhombus of uniform rods is suspended by one angle which is connected with the opposite angle by an elastic string of modulus the weight of the rhombus and of natural length that of one of its sides. The rhombus being supposed to remain in a vertical plane with the string vertical, find (a) the position of equilibrium, (b) the kinetic energy when the lowest point is moving with velocity v, (c) the time of a small oscillation about the equilibrium position.