(iv) Find the equation of that common chord of S1 and S2 which does not pass through either A or B.

9. A particle of weight W is held at rest on a rough plane inclined to the horizontal at a fixed (acute) angle a by a light string which makes an angle B (<π-α) with an upward line of greatest slope in the plane. The angle of friction between the particle and the plane is A. If the particle is just on the point of slipping up the plane, find an expression for the tension T in the string in terms of W, X, α and ß.

Find the greatest and least values of the tension T for varying 8 in the range 0≤ß≤π-α, distinguishing carefully the cases which arise for different given values of a and λ.

α

10. A uniform solid cylinder of radius a is projected in a direction perpendicular to its axis along a rough horizontal table. The initial velocity of the axis of the cylinder is V and the cylinder has a backward spin of magnitude N. Show that after slipping has ceased the cylinder rolls on the table with constant velocity (2V-aN)/3.

Describe in general terms the motion in the three cases 2V >aQ, 2V =ɑN, 2V <aN.

Show that if the cylinder has just returned to its startingpoint at the moment when slipping ceases, then 5V =aN.

11. A bead of mass m moves under gravity on a smooth wire bent into an arc of the cycloid

x=a (0-sin 0), y=a (1-cos )

(0≤0≤2π),

where the axis of x is horizontal and the axis of y points vertically downwards. The bead is released from rest at a point whose distance from the lowest point N of the wire, measured along the arc, is c. Find the speed with which the bead reaches N, and show that the reaction of the wire on the bead is then

12. Two equal particles P1, P2, each of mass m, are joined by a light inextensible string of length 21. The particles initially lie at rest on a smooth horizontal table with the string taut. At time t=0 the particle P1 is given a velocity V at right angles to the string. Find expressions for the speeds of the particles after time t.

When t=2πl/V the string comes into contact with a smooth peg fixed on the table. At the moment of contact the distance between P, and the peg is a. If during the subsequent motion the distance between P, and the peg is denoted by r, prove that for 2ml < Vt <2πl +√ 812 — 2a2

r2 = a2 + (Vt - 2πl)2.

A 21/I

OXFORD LOCAL EXAMINATIONS

GENERAL CERTIFICATE OF EDUCATION

Summer Examination, 1958

Advanced and Scholarship Level

PHYSICS, PAPER I

MONDAY, JUNE 23. TIME ALLOWED-3 HOURS

[Write the number of the paper, A 21/1, on the left at the head of each sheet of your answers in the space provided.

Answer SIX questions only, including at least ONE question from each of the four sections of the paper. Mathematical tables are provided.]

[Take g to be 981 cm./sec.2]

SECTION A

1. Explain how the principle of Archimedes is applied to the determination of densities.

A cube of tungsten carbide alloy with 9 cm. sides weighs 10.00 kg. in air, and 0.20 kg. when totally immersed in mercury. Find the density of the alloy and that of mercury.

Assuming that brass weights were used, find the airbuoyancy correction for the first of these weighings.

[Take the density of brass to be 8.4 gm./c.c., and that of air to be 0.00126 gm./c.c.]

2. Define moment of inertia. Explain why a flywheel is made with most of its mass concentrated near the rim, and discuss briefly the part played by the flywheel in a single-cylinder engine.

A small flywheel is on an axle 3 mm. in diameter. The moment of inertia of the system about its axis is 3,000 gm. cm.2, and it is set in motion by applying ́a steady tension of 100 gm. wt. to a thin thread wound round the axle. Assuming that the thread does not slip on the axle, and that there is no friction between the axle and its bearings, calculate (a) the angular acceleration while the thread is being pulled, (b) the angular velocity acquired by the flywheel when 50 cm. of thread has been pulled off the axle.

3. State the law of conservation of momentum, and describe how you would use a ballistic balance to demonstrate the law.

The two colliding bodies of a ballistic balance are hung by cords 2 metres long, and are arranged to stick together after impact. The body A, of mass 500 gm., is initially at rest; the other, B, of mass 700 gm., is released from rest at a point which is at a horizontal distance of 50 cm. from the point of collision. Find (a) the velocity of B before impact, (b) the common velocity after impact, and (c) the kinetic energy lost at the impact.

SECTION B

4. Explain the principles underlying the establishment of a scale of temperature, and discuss the meaning of the phrase 't° C. on the constant volume gas meter scale'.

thermo

Describe the constant volume gas thermometer, and state its advantages and disadvantages as a practical instrument for temperature measurement.

Assuming that real gases obey van der Waals' equation

a

(P+)(V-b) = RT,

where, for the individual gas concerned, a and b are constants and T denotes temperature on the ideal gas scale, show that readings on the constant volume gas scale for any gas should agree with those on the ideal gas scale.

5. State Newton's law of cooling, and describe how you would test it experimentally.

Discuss the factors that determine the rate of fall of temperature of a hot liquid contained in a calorimeter exposed to the air.

A calorimeter, of water equivalent 20 gm. and containing 150 c.c. of water, cools from 60° C. to 40° C. in 7 minutes. When the water is replaced by 150 c.c. of oil of specific gravity 0.8, the same calorimeter cools through the same temperature range in 4 minutes. Calculate the specific heat of the oil.

6. From the point of view of the kinetic theory, discuss the state of equilibrium between a liquid and its saturated vapour, and explain why the saturated vapour pressure of a liquid increases as the temperature rises.

Describe an experiment by which the saturated vapour pressure of water over the range 60° C. to 110° C. may be measured.

In Wollaston's cryophorus (Fig. 1), the sealed system contains water and water vapour only. When the bulb A is surrounded by a mixture of ice and salt, the water in B is eventually frozen. Explain this.

A

FIG. 1.

SECTION C

7. Explain, in terms of the wave theory of light, how a converging lens brings a plane wave-front to a focus.

A small object on the axis of a thin lens is at distance x from the first principal focus of the lens; its image is at distance y from the second principal focus of the lens. Prove from first principles that |xy| = f2, where f is the focal length of the lens.

Describe how you would determine the focal length of a lens experimentally by a method involving the use of the above relation.