# University of St Andrews Natural Philosophy Examinations 1884-5

At the University of St Andrews two Natural Philosophy papers were set in October 1884 and two papers on the same topics in April 1884.

In addtition two papers in Honours Natural Philosophy were set in 1885.

#### NATURAL PHILOSOPHY: OCTOBER 1884. First Paper.

Statics and dynamics.
1. Enunciate the proposition called the polygon of forces. Deduce it from the parallelogram of forces.
2. If any number of forces in one plane act at a point, and if their resolved parts in two directions vanish, prove that the forces are in equilibrium.
3. Find the magnitude and line of action of the resultant of two parallel forces acting on a body in the same direction.
4. Prove that if two forces pass through a point, the moments of the two forces about any other point are equal to the moment of their resultant about that point.
5. Find the centre of gravity of a triangle
6. Given the base and height of a triangle, show how to determine its shape that it may just rest in equilibrium when placed with its base on a horizontal plane.
7. Prove that if a body start from rest under the influence of a uniform acceleration $f$, it will in a time $t$ have traversed a space $\large\frac{1}{2}\normalsize f t^{2}$.
8. Show that the time which a body takes to slide down a chord of a vertical circle starting from rest at the highest point is the same for all chords.
9. Prove that the path of a projectile in a vacuum is a parabola.
10. Show that with a given velocity the greatest range on a horizontal plane is got when the direction of projecture makes 45° with the vertical.

#### NATURAL PHILOSOPHY: OCTOBER 1884. Second Paper.

1. Explain how the specific gravity of a liquid can be obtained.
2. What are the laws of the reflection and refraction of light? Explain the law of reflection on the undulatory theory of light.
3. How can the focal length of a convex lens be determined by experiment? How can the focal length of a concave lens be determined by experiment?
4. Describe the construction and method of using the gold-leaf electroscope.
5. What is meant by Ohm's law? If a given battery of internal resistance one ohm sends a certain current through a wire of eight ohms' resistance, what current will the same battery send through a wire of the same material of one half the diameter and three times the length?
6. What are the dark lines in the solar spectrum supposed to be due to?
7. What is the mechanical equivalent of heat? How can it be determined?
8. Explain what is meant by polarised light; and how polarised light may be obtained and detected.
9. Describe the formation of dew, and why it appears more readily on some objects than on others.
10. How can the velocity of light be determined?
11. How can the interference of two sounds so as to create silence be shown?
12. A circular wire is hung up, and another circular wire carrying an electric current is gradually approached to it in a parallel plane: describe what happens.
13. Explain what is meant by the Conservation of Energy.

#### NATURAL PHILOSOPHY: APRIL 1885. First Paper.

1. If forces $P, Q$, and $R$ act at a point $O$, prove-
$\Large\frac P{\sin QOR}\normalsize = \Large\frac Q{sin POR}\normalsize = \Large\frac R{\sin POQ}$.
2. Find the centre of gravity of particles weighing 1, 2, and 3 lb. placed at the angular points of an equilateral triangle.
3. Find the resultant of two unequal parallel forces acting in opposite directions.

Hence prove that a "couple" cannot be replaced by any single finite force acting at a finite distance.
4. Give Newton's laws of motion, and illustrate them by examples.
5. A ball (elasticity $e$) impinges obliquely on a smooth fixed plane: find its direction and velocity after impact.
6. By what means is the velocity produced by gravity reduced in Attwood's machine?
7. Define a simple pendulum and a compound pendulum.

How may the length of a simple pendulum and its time of vibration be determined by observations made with a compound pendulum? (Kater's method.)

The length of a seconds pendulum at a certain place is 39.15 inches: find the local value of $g$.
8. "In a machine," it is said, "what is gained in power is lost in speed:" explain and illustrate this.
9. Describe a method by which the specific gravity of a liquid may be determined.

A piece of brass weighing 1000 grammes weighs when immersed in water 880 grammes, and in alcohol 905 grammes: find the specific gravity of the alcohol.
10. What are the coefficients of linear and cubical expansion of a body?
How are they related numerically?
Why is a platinum wire used to fuse into glass?

#### NATURAL PHILOSOPHY: APRIL 1885. Second Paper.

(Not more than eighteen questions to be answered: a high value will be attached to the problems.)
1. In what three particulars may two notes differ? State the physical cause of each.
2. Describe any instrument by which the number of vibrations (per second) in a note of given pitch may be counted.
3. The report of a gun is heard 15 seconds after the flash is seen: find approximately the distance of the gun from the observer.
4. An open and a stopped organ-pipe are of equal lengths what is the difference in pitch between the notes they produce?
5. State the law of reflection from a plane mirror.

If a beam of light reflected from a plane mirror revolve through an angle of 30°, through what angle has the mirror turned?
6. What is meant by saying that the index of refraction of water is $\large\frac{4}{3}\normalsize$?
7. How was the velocity of light first measured?
8. Give a short description of the solar spectrum, and mention some of the principal lessons that have been learnt from the study of it.
9. Distinguish between conduction and convection of heat.
10. Heat near the equator, cold near the poles, and the earth's motion of rotation, may serve to explain the origin and general direction of the great ocean currents. Apply this to the Gulf Stream.
11. Show by a diagram the changes of volume of water substance under the influence of heat between the temperatures of (say) 5° and 105° Cent.
12. State the relation between the radiating and absorbing powers of a body. Give an illustration.
13. Define the capacity of a conductor: how may it be increased? Illustrate by describing the Leyden jar.
14. Describe the Electrophorus, and explain its action.
15. State Ohm's Law. How should the cells of a battery be arranged (i) when the external resistance is great compared with the internal, and (ii) when it is small?
16. Explain briefly how the currents are produced in the secondary coil of an induction coil.
17. If a circuit be acted on so as to increase the number of lines of force which pass through it, what effect is produced?
18. A tuning-fork makes 256 vibrations per second, and the velocity of sound is 340 metres per second: what is the wave length of the note produced?
19. A gas jet 16 feet and a candle 4 feet from a photometer illuminate it equally: compare the quantities of light emitted from the two sources.
20. If the refractive index of a ray of light in passing from air to water be and in passing from air to glass 3, find what it is for passing from water to glass.
21. A candle flame is placed at a distance of 3 feet from a concave mirror formed of a portion of a sphere, the diameter of which is 3 feet: find the position of the image formed by the mirror.
22. The combustion of 1 lb. of coal raises the temperature of 1000 lb. of water 4.4 degrees Cent.: find the mechanical equivalent of the heat produced by the coal.
23. The specific heat of iron is 0.113: how many lb. of iron at 250° Cent. must be used to melt 2 lb. of ice?
24. A Grove cell (E.M.F. 1.9 volts) is used to send a current through an external resistance of 100 ohms; find in ampères the strength of the current if the internal resistance of the cell is .25 ohm.
25. Compare the currents which the same electromotive force is capable of producing in two wires of the same material whose lengths are as 5 to 1, and cross sections as 3 to 2.

#### HONOURS NATURAL PHILOSOPHY: 1884. First Paper.

1. What is meant by absolute zero in the air thermometer?
2. Explain what is meant by the indicator diagram of a steam-engine.
3. What is meant by an adiabatic curve? Prove that where any adiabatic line crosses an isothermal line, it is inclined at a greater angle to the horizontal than the isothermal line.
4. Show how to indicate by a diagram the heat required to transform a body by any path from one state to another. Explain the meaning of the word path here.
5. Explain Thomson's absolute scale of temperature.
6. Explain Carnot's perfectly reversible engine, and explain that the efficiency of such an engine is the greatest that can be obtained with a given gauge of temperature.
7. Explain distinctly the meaning of the various symbols in the equation embodying the first law of thermodynamics - viz.:
$\Large\frac{dp}{dt}\normalsize = J\Large (\frac{dM}{dt}\normalsize - \Large\frac{dN}{dv} )$,
and deduce this equation.
8. Deduce the expression of the second law of thermodynamics in the form $\Large\frac {dp}{dt}\normalsize = J.M.C$, and explain the meaning of the symbols.
9. Prove that the work done by a perfect gas in changing from volume $v$ and pressure $p$ to volume $v'$ and pressure $p'$, without changing temperature, is
$pv \log \Large\frac{p}{p'}$.
10. Prove that if in a perfect gas the total quantity of heat be expressed as a function of $p$ and $v$, the pressure and volume,
$\Large\frac{d\phi}{dv}\normalsize = Ap$, and $\Large\frac{d\phi}{dp}\normalsize = Bv$.
Find expressions for $A$ and $B$.

#### HONOURS NATURAL PHILOSOPHY: 1884. Second Paper.

1. When a system of forces is reducible to a single resultant, find the equations to the line in which that resultant acts.
2. Show how to find the centre of gravity of a piece of a surface of revolution cut off by two planes perpendicular to the axis.

Show that in the case of a sphere it lies half-way between the two planes.
3. A body moves in an ellipse about a centre of force in the focus: when it arrives at the extremity of the minor axis, the law of force changes to that of the direct distance without the actual intensity of the force at the distance in question being changed : compare the time of revolution in the new orbit with that in the old orbit.
4. Prove that if an equipotential surface belonging to any electrical system be drawn, and a distribution of electricity be made over that surface such that the density at each point is $F/(4\pi)$, where $F$ is the resultant force of the system at that point, then this electrification will be in equilibrium, and will produce on all external particles the same force as the given electrical system.
5. Prove that the energy exerted in charging a conductor with $Q$ units of electricity to a potential $V$ is $\frac 1 2 QV$.
6. Prove that if two parallel conducting plates distant t from one another be respectively at potentials $V_{1}$ and $V_{2}$ , and if S be the amount of the surface of each, and $F$ the total mechanical force urging the conductors, then
$(V_{1} - V_{2})^{2} S = t^{2}.\delta\pi.F.$
7. Find the law of density on a freely electrified spheroid, and thence deduce that for an electrified disc of radius $a$, if ρ0 be the density at the centre, the density at a distance $x$ from the centre is
$\Large\frac {a\rho_0}{\sqrt{a^2 - x^2}}$.
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