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The beam of a balance is 25 cms. long and the pointer 30 cms. The knife edges are in one plane and the pointer-scale is divided to millimetres. If the weight of the beam is 200 grms. and 1 mgrm. moves the pointer through one division of the scale, shew that the distance of the centre of mass of the beam below the knife edges is 3/16 of a millimetre.

7. State and prove Archimedes' Theorem of Buoyancy.

The lower end of a vertical tube of radius r, which is immersed to a depth h in water, is closed by a float in the shape of a solid right circular cone of height k and radius of base a with its vertex uppermost. If the sp. gr. of the float is 5, find its pressure on the tube (supposed free of water).

8. The cross-sections of the two vertical limbs of a U-tube, one closed, the other open, are of areas A, a respectively. A volume V of air at atmospheric pressure is confined in the tube by mercury in the bend. If an additional volume v of mercury is poured into the tube, find its rise in the closed limb when the temperature is the same as before.

9. Prove the formula t = (p-II)r for the circumferential tension in a thin cylindrical pipe under internal and external pressures p, II.

Twenty cubic feet of gas, at a pressure of 30 inches of mercury, are compressed into onesixth of a cubic foot in a cylinder of 2 inches internal radius. Find the pressure in lbs. per square inch, and the necessary thickness of the cylinder, the permissible stress being 10 tons per square inch.

MIXED MATHEMATICS.-PART II.

FIRST PAPer.

The Board of Examiners.

1. State and prove a formula for the distance between two points on the (spherical) earth of latitudes X, X' and longitudes 1, l'.

If the points are in the same latitude λ, obtain a formula for the difference between their distances apart along a great circle, and along the parallel of latitude.

2. Prove the formula

cot a sin ccot A sin B + cos c cos B for any spherical triangle ABC.

A ship's course, after making an (angular) distance c on a great-circle, is the same as at the start. Shew that the latitude at either end is given by

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where is the inclination of the course to the meridian at either end.

3. ABC is a triangle right-angled at C. State and prove a formula connecting A, B, a.

Investigate the fundamental formulæ for a triangle of which one side is a quadrant.

4. State the adjustments of the transit circle, and give one method, with formulæ, of determining the magnitude of each of the errors of adjustment.

5. Find the alteration in altitude and azimuth of a given star due to a small change of position (n miles at an angle 0 E. of N.) of the observer.

6. Discuss the equation of time.

Find the R. A. and Declination of the mean sun in the ecliptic at a given time.

7. Describe, with as much mathematical detail as possible, a method of determining the mean distance of the moon from the earth.

MIXED MATHEMATICS.-PART II.

SECOND PAPER.

The Board of Examiners.

1. Investigate the theory of the centrodes in uniplanar kinematics.

One point of a disc moves with uniform angular velocity w in a circle of radius r, and the disc itself rotates with uniform angular velocity w'. Find the centrodes of the motion.

2. Find an expression for the kinetic energy of a rigid body rotating around a fixed axis.

Find the kinetic energy of a solid right circular cylinder of mass M and radius r rotating around its axis of figure with angular velocity w.

3. Prove the equable description of area in a central orbit.

A planet describing an orbit of major axis 2a and eccentricity e, receives an impulse when at aphelion, so that its velocity there is increased by 1/nth. Find the new periodic time.

4. Define the moment of a force about a line, and show that the application of the methods of reduction in Statics does not alter the sum of the moments of a system of forces about any line.

Find the couple required to hold a door, with oblique line of hinges, at a given angle with its equilibrium position.

5. Find the centre of mass of a segment of a solid sphere.

6. Enunciate the principle of virtual work and apply it to demonstrate the conditions of resolution and moments for the equilibrium of a rigid body under uniplanar forces.

7. Find the magnitude and line of action of the resultant pressure on a vertical rectangular area in heavy liquid, one diagonal of the rectangle being horizontal and at a given depth below the surface.

8. Demonstrate the stability of a rigid body, floating in liquid, for vertical displacement without rotation, and find an approximate formula for the time of a small oscillation.

MIXED MATHEMATICS.-PART III.

The Board of Examiners.

1. Find the intensity, pitch, and axis of the wrench equivalent to a system of forces such as (X, Y, Z) at (x, y, z).

A body under such a system of forces rests in contact with the two smooth planes + my + nz = p, l'x + m'y +n'z = p' at the points (X1, Y1, Z1) (X2, Y, Z). Find the analytical conditions of equilibrium and the pressures on the planes.

2. Investigate the general formulæ for the centre of mass of a solid in spherical polar coordinates.

Find the centre of mass of the part between z = 0 and z = h of a helicoidal solid, whose axis is the axis of z and whose pitch is p, (that is, the section by the plane z is the section by z = 0 turned through an angle z/p). Shew that the distance of the c.m. from the axis is

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where is the distance of the c.m. of the section z from the axis.

3. Investigate general equations of equilibrium for a string under any forces.

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