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CHAPTER I

DYNAMICS

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).

In examples on change of units, the following (approximate) relations may be assumed

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 " as 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.

5. A force of 100 dynes acts upon a mass of grammes for 5 seconds: what velocity does it genera

6. A constant force acting upon a mass of 30 gram causes it to move through 10 metres in 3 seconds, st ing from rest: what is the value of the force in dynes

7. A force of 1,000,000 dynes acts upon a body 10 seconds, and gives it a velocity of a metre per secon find the mass of the body in grammes.

8. How long must a force of 5 units act upon a bo in order to give it a momentum of 3000 units? (T unit of momentum is that of a gramme moving at t rate of one centimetre per second.)

9. During what time must a constant force of dynes act upon a kilogramme in order to generate in a velocity of 3 metres per second?

10. What force acting upon a mass of 50 gramm for one minute will produce a velocity of 45 centimetr per second?

11. A body moving with uniform velocity in a circ is commonly said to be acted on by "centrifugal force Discuss the correctness of this expression, stating whethe the quantity referred to is really a force. Is its actio centrifugal ?

12. State and explain Newton's Third Law of Motion and give examples of its application. If the earth at tracts 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 sho is 14 lbs. The shot leaves the gun with a velocity o 800 feet per second: what is the initial velocity of the recoil?

14. A 56-lb. 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.

15. Do you consider weight to be an 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-oz. 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,

24. A body of mass 6 lbs. is acted upon by a f of 30 poundals: find its velocity and momentum at end of half a minute.

25. A spring-balance is carried in a balloon whic ascending vertically. Find the acceleration of the ball when a half-pound weight hung upon the spring-bala is found to indicate 9 ounces.

26. By what number would the acceleration du gravity be expressed, if the day and the mile were units of time and length?

27. If the unit of length were the yard, the uni time the minute, and the unit of mass a mass of 10 1 what would be the value of the unit of force in ter of the poundal?

28. A certain force acts upon a mass of 150 gramm for 10 seconds, and produces in it a velocity of metres per second: compare the force with the wei of a gramme.

29. A force equal to the weight of one pound a upon a ton: what acceleration is produced, and w will be the velocity at the end of 10 seconds?

30. A certain force acts upon a mass m and genera in it an acceleration a: find the weight which the fo would statically support.

31. A spring-balance is graduated at a place whe g=32.3; at another place where g= 32, a body is test and the balance indicates 8 oz.: what is the corr mass of the body?

32. A certain force acting on a mass of 14 poun for 10 seconds generates in it a velocity of 128 feet 1 second. Compare the force with the weight of o pound, and determine the acceleration which it wou produce in a mass of one ton.

33. An 18-ton truck is moving at the rate of miles per hour: what is its momentum? (Take t foot and pound as the units of length and mass.)

34. Compare the momentum of a 15-lb. cannon-b

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moving at the rate of 300 feet per second, with that of a 3-oz. bullet which has a velocity of 700 yards per second.

35. In what time will a velocity of 45 miles per hour be generated in a train of 80 tons, if the engine exerts upon it a pull equal to a weight of 2 tons?

36. If a body of mass 10 kilogrammes be acted upon for one minute by a force which can statically support 125 grammes, what momentum will it acquire?

37. A body of mass 4 lbs. is observed to be moving at a rate of 8 feet per second; at this instant a constant force begins to act upon it in the direction of its motion, and after 20 seconds its velocity has increased to 24 feet per second. Determine the magnitude of the force, and explain clearly what unit of force you employ in your solution.

38. Compare the amounts of momentum in (1) a 56-lb. weight which has fallen for 2 seconds from rest, and (2) a cannon-ball of 12 lbs. moving with a velocity of 900 feet per second.

39. If the mile be taken as the unit of length, and the acceleration caused by gravity as the unit of acceleration, what will be the unit of velocity?

40. A 7-lb. weight hanging over the edge of a smooth table drags a mass of 56 lbs. along it: find the acceleration, and the distance moved through in 5 seconds from rest.

41. A falling weight of 162 grammes is connected by a string to a mass of 1800 grammes lying on a smooth flat table: find the acceleration, and the tension of the string.

42. A mass of 3 lbs. is drawn along a smooth horizontal table by a mass of 6 oz. hanging vertically : calculate the space described in 3 seconds.

43. A force of 30 dynes acts for 12 seconds upon a body resting on a smooth horizontal plane, and imparts to it a velocity of 120 centimetres per second what is the mass of the body?