59. We have defined force as that which when applied to mass produces acceleration. No force however great can produce in a mass a finite addition of velocity unless the force acts (producing acceleration in the mass) throughout a finite interval of time. 60. DEF. An impulse is that which acting on a mass produces in it a finite velocity, so that the massvelocity produced varies as the impulse. 61. DEF. We shall choose as our unit impulse that impulse which acting on a pound produces in it i velo. We shall call the unit impulse a pulse. It should be noticed that in the definition of an impulse nothing is said about the length of the interval occupied by the impulse. An impulse is quite a different thing from a force—just as different as a velo is from a celo. Examples. Each entire stroke of a tricyclist's foot on the pedals is an impulse. Each complete down stroke of the handle of a pump is an impulse. 62. An impulse is the equivalent of a force acting for some interval. A pulse has the same effect on a mass as a poundal acting throughout i second; each produces a pound-velo in the mass on which it has acted. It has the same effect as (i) 2 poundals acting for second; (ii) 100 poundals acting for Tooth of a second; (iii) 10000 poundals acting for yoooth of a second; and SO on. 63. When a mass has received a finite addition to its velocity we may say either (i) that it has received an impulse; or (ii) that a force has acted on it for a finite interval. Example. A shot of 8 cwt. from a 38 ton gun is observed to leave the gun with a velocity of 2000 velos; (i) find the impulse it has received; (ii) if the shot passes over 10 ft. when in the tube of the gun, find the resultant force which has acted on the base of the shot, supposing that force to have been uniform. The shot has received 2000 x 8X112 pound-velos. (i) The force which produced this mass-velocity was uniform ; so also therefore must have been the acceleration; also the shot passed over jo ft. while in the gun. Let the acceleration be a celos, then, (Art. 26, iii] a x10=12*=* (2000)”, whence, The mass of the shot is 8X112 lbs.; so that its mass acceleration is 8 x 112 x 200000 pound celos. To produce this, 8x112 X 200000 poundals are necessary. The force is therefore 179200000 poundals, equivalent to the weight of 179200000 tons; that is, of 3124 tons. (ii) 32 X 20 XII 2 a= 200000. *64. The effect of a large force acting during a short interval and that of a small force acting for a large interval may each be equal to the effect of the same impulse. Thus an impulse requires two things to produce it (i) an interval of time, (ii) a force. A pulse might be termed a poundal-second ; meaning that a pulse has the same effect as a poundal acting for a second. Suppose the interval occupied by 9 pulses is t seconds; 9 pulses produce a pound velos. Also, poundals acting throughout t seconds produce a pound velos; therefore, the average force which produces a pulses in t seconds is poundals. Hence, the measure of the average force which produces a given impulse is the ratio of the number of pulses acting to the number of seconds required. In other words the average force producing an impulse is measured by the rate of impulse per second. Example. A glass ball of 4 oz. strikes a pavement with 50 velos, and is observed to rebound with 30 velos; find the impulse which it receives ; supposing the impulse occupies do th of a second, find the average magnitude of the stress between the ball and the pavement. [Note. When a glass ball is let fall upon a horizontal pavement the velocity on reaching the pavement seems to be very quickly changed. The motion of the pavement itself is imperceptible (for a reason explained in Art. 73) and is neglected. Here, the velocity on reaching the pavement is 50 velos ; when that velocity ceases, the ball has received 50 velos upwards ;-for, 50 velos downwards and 50 velos upwards produce rest ;—when the ball is seen moving upwards with 30 velos, it has received 30 velos more; so that in all it has received 80 velos upwards. These velos were produced by a force, which produced acceleration upwards, and which acted for some interval; the interval was very small (too small perhaps to be observed) but still an interval. The interval given in the question viz. ttoth of a second is too small to be actually observed.] The change in the velocity of the ball is 80 velos; therefore the mass-velocity produced is 18 x 80 pound-velos; that is, 20 pound-velos, and this requires an impulse of 20 pulses. Let x be the average number of poundals in the stress; then, x poundals acting on 4 lb. produce 4x celos, 4x celos in ito secs. produce pound-velos; 4X hence, Іоо 4.3 = 20; IOO x=500. The average force on the ball is 500 poundals; that is, the weight of 50 pounds; that is, of 15% lbs. or, EXAMPLES. XVIII. 1. A shot of 8 cwt. leaves a fixed 40 ton gun with a muzzle velocity of 2000 ft. per second ; find the impulse which has acted upon it. Also if the shot moves over 20 ft. when in the gun, find the resultant uniform push which has acted on the base of the shot. 2. Find how long the shot in question 1 was in motion in the tube of the gun. 3. On Friday Feb. urth, 1886 at Woolwich Arsenal a gun of un tons was fired for the first time with an iron shot of 1800 lbs. ; the muzzle velocity was 1600 velos; find the impulse which acted on the shot. Also if the shot moved over 34 ft. while in the gun find the resultant uniform push which was applied to its base to produce this velocity. 4. Find how long the shot in Question 3 was in motion in the tube of the gun. 5. A glass ball of 1 lb. reaches a horizontal pavement with 20 velos and leaves it with 12 velos both velocities being vertical; what impulse has acted on the ball ? and supposing the ball and pavement were in contact for to sec., find the average external force acting on the ball. 6. A glass ball of 8 oz. is let fall from a height of 16 ft. and is observed after rebounding to rise to the height of 4 ft., find the impulse. 7. Two equal railway carriages of 3 tons moving in opposite directions each with a velocity of 10 miles per hour meet, and rebound each with a velocity of 5 miles an hour; find the impulse which has acted on each. Find also in tons weight the force between the buffers, supposed uniform, assuming that they were in contact for a second. 8. An india-rubber ball weighing 1 lb. strikes a pavement at right angles with a velocity of 30 velos, and rebounds with a velocity of 25 velos; find the impulse which has acted upon it; and find what force acting for both of a second would have the same effect. 9. A steam hammer of 10 tons is let fall from the height of 4 st., and strikes a mass of red-hot iron; supposing that the hammer on touching the iron comes to rest in zand of a second, and that the force it exerts is uniform, find the push, and by how much it flattens the red hot iron. 10. A hammer of 2 lbs. hits a nail with a velocity of 40 ft. per second and drives the nail in half an inch ; find (i) the force which is exerted on the nail supposing it uniform, (ii) the duration of the impulse. 11. An iron weight of half a ton is let fall from a height of 16 ft. vertically above the head of a pile ; the pile is driven into the ground a distance of 1 ft. by the impulse; find (i) the average force, (ii) the duration of the impulse. * 65. When one mass meets another mass having a different velocity the masses are said to impinge. The meeting of the two masses is called an impact. The impact is said to be direct when the masses are moving in the same straight line, provided the direction of the stress caused by the impact is also in that straight line. Let us consider the case of one mass having a certain velocity impinging on another mass having a different velocity, the masses being each composed of a hard substance, such as a metal or glass. In one or both of the masses there must be a very rapid change of velocity ; that is, there must be a very great acceleration; consequently there must be an immense mutual pressure between the masses. In such cases the impact must certainly occupy some interval of time : the particles of each mass yield slightly in the neighbourhood of the great pressure, and then, either return more or less exactly to their original position (in which case the material is said to be elastic), or do not return (in which case the impact causes a permanent indentation, and the material is said to be inelastic). 66. It should be noticed [Art. 62] that the smaller the interval occupied by a given impulse, the larger is the average force which produces that impulse. Also, the smaller the interval, the smaller is the distance passed over by the mass during the impulse. For, let m lbs. moving with u velos have mv pulses applied to it in t seconds; then, assuming that the v velos are produced uniformly, s=(u+ }v) xt; so that the smaller t, the smaller is s. 67. An impact between two masses causes a stress to be set up between them; which stress lasts for a small interval. The action and reaction of this stress produce equal external impulses which act on the two masses in opposite directions. |