manner in which these operations are carried out, and the method of finding a number when its logarithm is given, will be best explained by an example. 453.6 x 30.48 by the use of fourplace logarithms. Ex. 7. To find the value of (2:54)2 x 1 3600 log 453.6 = 2.6567 log (2:54) = 2 log 2.54 = 2 x 0.4048=0.8096 log 13600 = 4:1335 log dividend = 4.1407 =4.9431 log quotient =ī.1976 log divisor = 4.9431 0.1976 is not one of the logs given in the table : the next lower one is 0.1959, which is the log of 1:57. Now 0.1976 .-0.1959 =0.0017. Looking along the row (in which the log is given) for 17, we find that it stands in a column headed by the figure 6, and this is the fourth figure of the number. Lastly, by rule (2), we see that 1.1976=log 0.1576. The value of the fraction is therefore o.1576. A full account of the methods of logarithmic calculation will be found in Chambers's Mathematical Tables, and these may be used for more accurate work ; but the table of four-place logarithms at the end of this book will be found sufficient for working out most of the problems given. The student is advised to practise the methods of approximation indicated in § 11, and to make himself thoroughly familiar with the use of logarithms, as a large amount of arithmetical calculation will thus be avoided. In working out examples he should aim at something more than merely getting a correct numerical answer : diagrams or rough sketches should be given wherever they render the solution more intelligible, and formulæ should not be quoted without explanation unless the relations which they express are perfectly well known and easily remembered. In particular, every step in the reasoning should be carefully thought out and clearly explained, for the solution of problems is not so much an end itself as a means of acquiring thorough and intimate acquaintance with physical laws. CHAPTER I DYNAMICS C.c. Note.—In all the examples, excepting where otherwise stated, the numerical value of g is taken as 981 when the centimetre and second are the units of length and time, and as 32 when the foot is the unit of length. The abbreviation cm. is used for centimetre(s). gm. gramme(s). cubic centimetre(s). In examples on change of units, the following (approximate) relations may be assumed I foot 30.48 cm. I inch 2.54 cm. I pound = 453.6 gm. 1. State and discuss Newton's First Law of Motion, and show that it provides us with a definition of force. 2. Enunciate Newton's Second Law; state the exact meaning of the phrase change of motion used by him, and explain how the law enables us to measure forces. 3. Starting from Newton's Second Law of Motion, show how to obtain a definition of the C.G.S. unit of force (the dyne). If the weight of a gramme be taken as the unit of force, what is the unit of mass ? 4. A force of 25 units acts upon a mass 10: find the acceleration produced, and the space described in 30 seconds from rest. as 5. A force of 100 dynes acts upon a mass of 25 grammes for 5 seconds : what velocity does it generate ? 6. A constant force acting upon a mass of 30 grammes causes it to move through to metres in 3 seconds, starting from rest : what is the value of the force in dynes ? 7. A force of 1,000,000 dynes acts upon a body for 10 seconds, and gives it a velocity of a metre per second : find the mass of the body in grammes. 8. How long must a force of 5 units act upon a body in order to give it a momentum of 3000 units ? (The unit of momentum is that of a gramme moving at the rate of one centimetre per second.) 9. During what time must a constant force of 60 dynes act upon a kilogramme in order to generate in it a velocity of 3 metres per second ? 10. What force acting upon a mass of 50 grammes for one minute will produce a velocity of 45 centimetres per second ? 11. A body moving with uniform velocity in a circle is commonly said to be acted on by “ centrifugal force.” Discuss the correctness of this expression, stating whether the quantity referred to is really a force. Is its action centrifugal ? 12. State and explain Newton's Third Law of Motion, and give examples of its application. If the earth attracts the moon with a force F, what is the attraction exerted by the moon upon the earth ? 13. The mass of a gun is 2 tons, and that of the shot is 14 lbs. The shot leaves the gun with a velocity of 800 feet per second : what is the initial velocity of the recoil ? 14. A 56-1b. shot is projected with velocity v from a gun, the mass of which (together with its carriage) is 6 tons. Express, in terms of v, the velocity of recoil of the gun. an 15. Do you consider weight to be essential property of matter ? State clearly what distinction you would draw between mass and weight ; and illustrate your remarks by reference to the force required (1) to open a large iron gate, well balanced and swinging upon good hinges, and (2) to lift up the same gate when lying on the ground. 16. Explain what is meant by “the acceleration due to gravity.” If its numerical value be 32 when the unit of length is the foot and the unit of time the second, what is its value when the unit of length is the yard and the unit of time the minute ? (See & 8, Ex. 1.) 17. A 4-0z, weight is suspended from a spring-balance which is carried in a balloon; what will be its apparent weight as shown by the index (1) when the balloon is ascending with an uniform acceleration of 8 feet per second, (2) when it is descending with a velocity of 16 feet per second ? 18. What do you understand by the phrase “ weight of a pound”? The British unit of force (called a poundal) is defined as being that force which, acting upon a pound mass for one second, generates in it a velocity of one foot per second : how many poundals are there in a pound weight ? 19. Explain the distinction between gravitation measure and the absolute measure of force. Show also how the one may be expressed in terms of the other, finding, for example, the number of dynes in a gramme weight. 20. Express the weight of 10 kilogrammes in dynes, and the value of a dyne in terms of a gramme weight. 21. Calculate the value of a pound weight in dynes. (See note on p. 23.) 22. A force of 980 dynes acts vertically upwards upon a body of mass 5 grammes, at a place where g=981: find the acceleration of the body. 23. A force equal to a weight of 10 lbs. acts upon a mass of 25 lbs.: what is the acceleration produced, and what momentum will be generated in 5 minutes ? |