4. Make a careful drawing of a system of weightless and frictionless pulleys in which a force equal to the weight of one pound can support a weight of 31 pounds.

Show that, in the system you adopt, what is gained in power is lost in speed.

5. Make a careful drawing of the beam of a delicate balance (i. e. pair of scales) pointing out the conditions which must be secured by the maker in order to insure accuracy. How can the sensibility of such a balance be varied?

What difference in principle is there between the employment of a pair of scales and of a spring balance, and under what circumstances will the two methods lead to different practical results?

6. A cylindrical vessel one foot in diameter and one foot high, open at the top, is made of thin sheet metal of uniform thickness. Find the height of the centre of gravity from the bottom of the vessel. If the vessel be half filled with water, where will the common centre of gravity of the vessel and water be, assuming that the weight of the vessel is onefifth of the quantity of water in it.

7. A train of 200 tons is urged forwards with a force equal to the weight of 11⁄2 tons, while the resistance it experiences is equal to the weight of 1 ton. What will be the measure of its acceleration, and how long will it take to acquire a velocity of 10 miles an hour?

B.

8. Define Specific Gravity. A piece of lead and a piece of wood balance one another when weighed in air. Which will weigh the most, by how much, and why?

9. A cylindrical diving-bell weighs two tons and has an internal capacity of 200 cubic feet, while the volume of the material composing it is 20 cubic feet. The bell is made to sink by weights attached to it. At what depth may the weights be removed and the bell just not ascend,it being given that the mass of a cubic foot of water is 1,000 ozs. and the height of the water-barometer 33 feet?

10. The standard atmospheric pressure is that due to 29.905 inches of Mercury at the freezing point in the latitude of London and at the sea level. Explain fully this statement.

Supposing the atmospheric pressure to remain the same, how would the height of the barometric column be affected by carrying it to the equator, and why?

11. State the law connecting the volume, pressure and temperature of a given mass of gas. Descrtbe a method of investigating the relation between the pressure and temperature of a quantity of gas whose volume is kept constant.

12. Define specific heat.

Describe a method of determining the specific heat of a solid.

How many units of heat would cause a mixture of ice and water to contract by 50 cubic millimetres, if 100 cub. mm. of water at O°C. become 109 cub. mm. of ice on freezing?

13. What are the laws of the mixture of gases and vapours? Explain

the principle of, and describe the method of using, the wet and dry bulb hygrometer.

14. Explain fully why a stick plunged in water should appear bent. Make a careful drawing showing the path of a pencil of light from a point on the stick to the eye, and explain how the position of the visible image is determined.

15. Show how to find the relation between the dimensions of an object and its image formed by a convex spherical mirror.

Find the size and position of the image of a square an inch in diameter placed eight inches in front of a convex mirror of six inches radius.

16. How would you determine experimentally the focal length of a lens when parallel light is not available?

A lens whose focal length is two inches is employed as a magnifying glass. Where must the object be placed that its image may be eight inches from the lens? Draw carefully a figure showing the paths of pencils of rays from the object to the eye and the formation of the image.

ANSWERS TO THE NATURAL PHILOSOPHY PAPER.

1. A force is that which excites or tends to excite motion in a particle. Momentum is the quantity of motion residing in a body, and is measured by the product of the mass and the velocity.

Impulse is the force of a blow, and is measured by the momentum generated in the body struck.

Energy is the capacity for doing work.

(1) If the cannon-ball stop dead, its entire momentum is transferred to the target. Therefore the measure of the impulse is— 1,500,000 units.

mv. = 1,000 × 1,500

=

(2) If the cannon-ball rebound at the rate of 200 ft. per second, it must have lost 1,000 (1,500 + 200) units of momentum. This has been imparted to the target. Therefore the impulse is measured by 1,700,000.

2. If a particle has two independent velocities, Oa and Ob, when it starts from the point O, its resultant velocity will be measured by Oc, the diagonal of the parallelogram formed with Oa and Ob as adjacent sides. For at the end of one second the particle will have moved from 0 to b under the influence of its velocity Ob alone, and from b to c under the further influence of the velocity Oa. Since bc Oa, its resultant velocity then will carry it in one second from 0 to c, that is to say, along the diagonal Oc.

=

Since the two motions of the ship are north and east, they are at right angles.

The sailor's motion is at right angles to the plane in which the ship is moving, and therefore to the line of motion of the ship:

3. Vide text-books.

Let G be the centre of gravity, then the triangles ABG, BCG, CAG

are all equal.

Therefore the moments of the three forces about the centre of gravity are all equal and the board will turn about its centre of gravity.

4. In the third system of pullies, where the string which passes round any pulley is attached at one end of it to the weight, and at the other end to the next pulley, the strings being all parallel:

(2-1) P.

31. P=1

2n = 32.

n = 5.

Five pullies are therefore needed, one of which is fixed and four are moveable.

Call the four moveable pullies A4, A ̧, A2, A1. If the weight is raised through a space S the pulley A' will be lowered through S.

A, will be lowered through 2 S in consequence of A, being lowered through S, and through S besides in consequence of W being raised through S: i. e. As will be lowered through (2+1) S; similarly A, will be lowered through 2 (2 + 1) S + S, i. e. (22 + 2 + 1) S. And A, will be lowered through 2 (22+ 2 + 1) S + S, i. e. (23 + 22 + 2 + 1) S. And finally P will be lowered (21 + 23 + 22 + 2 + 1) S 31 S. And therefore one pound has to descend 31 feet in order to raise 31 pounds through one foot.

5. To insure accuracy in the balance—

(i) The two arms of the beam ought to be precisely equal. (ii) The balance ought to be in equilibrium when the scales are empty. (iii) The beam being horizontal, its centre of gravity ought to be in the same vertical line with the edge of the fulcrum, and a little below the latter.

The sensibility of a balance can be varied(i) By altering the lengths of the arms. delicate will be the balance.

The longer the arms,

the more

The lighter the arms, pro

(ii) By altering the weight of the arms. vided they are rigid, the more delicate the balance.

(iii) By raising the centre of gravity of the beam or lowering it, provided it is always below the point of support. The balance will be more delicate, according as the centre of gravity is nearer the point of support.

Owing to the fact that the force of gravity decreases from the poles to the equator, a given mass would deflect a spring less at the equator than in higher latitudes; consequently a body when weighed by a spring balance at the equator would appear too light. If weighed in a pair of scales, the standard weight would lose as much as the body to be weighed, and consequently the weight of the body would appear unaltered.

6. Let G be centre of gravity of cylinder, and K the centre of gravity of the base, and L the centre of gravity of the whole.

2hπr

Tr2
= 24 = 4.
GK:

=

Now let M be the centre of gravity of the water, and N the centre of gravity of the vessel and the water.

=

MN × 5w LN × w, or ML : LN:
inches: .. LN

=(4층 3) inches above the base.

=

=

6:5,

inches and N is 44 inches

8. The specific gravity of a body is the number which expresses the relation of a given volume of the body to the same volume of another substance taken as standard. Water is usually taken for the standard of solid and liquid bodies, air for the standard of gases.

A body when weighed in air losses a portion of its weight proportional to the weight of air which it displaces. Lead is denser than wood, and consequently for equal weights the volume of the wood is greater than that of the lead. Wood therefore displaces a larger quantity of air than lead, and the true weight of the wood will be greater than that of the lead by the weight of air whose volume is the excess of the volume of the wood over that of the lead.

9. Let S be the sp. gr. of the material of the bell.

20 cubic feet weigh 2 tons.
1 cubic foot weighs 2 cwt.

3.584. 1000

3584 oz.

If V is the volume of the bell, the force which presses it down is (S-W) V where W is the sp. gr. of water.

The force which presses it upwards is due to the buoyancy of the imprisoned air, and if D' is density of the air at the required depth and V' its volume, this force is (W-D')V'.

In order that the bell may just not rise these forces must be equal.

(S - W) V

=

(W-D')V'

or (3.5841) 20 V' - V'D',

=

but V'D' is a constant quantity and is equal to the product of volume and density of air under ordinary pressure

... V'

=

200 X 0.00125. = 51.93.

Now let x be the depth of the surface of the water in the bell.

Therefore the weights may be detached when the surface of the water in the bell has reached a depth of 94.09 feet.

10. The weight of the atmosphere varies at the same place at different times. But the average weight at 0° and sea-level in the latitude of London of a column of air whose height is that of the atmosphere, and whose base is the unit of surface, is the same as that of a column of mercury whose height is 29.905 inches, and whose base is the unit of surface.

The latitude musi be taken tnto account, because the force of gravity decreases from the poles to the equator.

The height above sea level must be taken into account, because, as we rise higher, the column of air above grows shorter, and the pressure consequently diminishes.

The temperature must be taken into account, because mercury expands with heat and grows specifically lighter, and therefore under the same pressure will stand higher at the Equator than at London, owing to the rise in temperature.

11. The volumes of gases are inversely proportional to the pressures to

V P'

which they are subjected, or v

=

P.

If the volume of the gas remains constant, suppose its volume at 0°, and pressure P is V., then for every degree in the rise of temperature the pressure must increase

Р

273.

12. The specific heat of a substance is the ratio to a thermal unit of the quantity of heat required to raise one gram mass of the substance from 0° to 1°.

The specific heat of a solid may be determined

(i) By its rate of cooling;

(ii) By the fusion of ice;

(iii) By the method of mixtures.

Full accounts of these methods are given in all text-books.

Let be the quantity of ice which on melting becomes x 50 cubic millimetres of water.

50 109 100

100 x = 109 x 5450

x=

605.5 cubic millimetres of ice.

This becomes 555.5 millimetres of water,

or 0.5 cubic centimetres of water.

The quantity of heat required to melt 1 gram of ice is 79 units.

.. The quantity of heat required to produce the given contraction is 79 × 0.5 = 42.8 units.

13, (i) The mixture takes place rapidly and is homogeneous.

(ii) If the gases severally and the mixture have the same temperature, and ten gases severally and the mixture occupy the same volume, then the pressure on the unit area exerted by the mixture will equal the sum of the pressures on the unit area exerted by the gases severally.

For a description of hygrometers and their method of use, vide any textbook, such as Ganot's Physics, § 386.