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EXAMPLES. XV.

1. A sledge weighing 1 ton is pulled on ice by a rope whose tension is equal to the weight of 56 lbs.; if the friction of the ice causes a horizontal retarding force on the sledge equal to ob of its weight find the acceleration produced.

2. A sledge, weighing 1 ton is pulled on ice by 4 dogs whose weight is 28 lbs. each; each dog obtains from the friction of the ice a horizontal force forwards equivalent to half its own weight; prove that if the friction of the sledge on the ice is of its weight the acceleration produced is 4% celos:

3. A man of 12 stone on a railway truck weighing 11 tons propels it, when he is himself on it, by means of a rope fastened to a fixed point; he pulls with a constant force of 40 lbs. weight; compare the acceleration of the truck with that which he would produce if, fastening the rope to the truck and getting off, he pulled at the rope with a force of 40 lbs. weight; supposing that in each case a force of 12 lbs. weight is required to overcome the friction etc.

4. A mass of i} tons placed on a smooth horizontal table is acted on by a horizontal force of 40 lbs. weight; find how far it would move from rest in 2 seconds.

5. A mass of 11 tons lies on a smooth horizontal table and is attached by a light string, which passes over the edge of the table, to a mass of 28 lbs. which hangs freely under the action of its own weight; how far will the masses move from rest in 2 seconds?

6. A man of 12 stone finds that he can in running attain a speed of 15 miles an hour in 3 seconds; find the horizontal force which he applies to his own mass, supposing that he attains that speed with uniform acceleration.

7. The man in Question 6 ties a string to a mass of 5 cwt. on a smooth horizontal sheet of ice and starts to run on the bank, also hori. zontal, using exactly the same horizontal force as in 6; how long will he take to attain a speed of 15 miles an hour?

8. A train of 240 tons running at the rate of 30 miles an hour on a straight horizontal line comes to rest, when under the action of friction only, after running 4 miles; assuming the force of friction to be constant, find the force in lbs. weight which would just cause the train to move when at rest.

9. Express in lbs. weight the force which the engine of the train of Question 8 must exert that it may attain a velocity of 60 miles an hour in i min. 28 secs. from rest.

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49. DEF. The inertia or inertness of mass is that essential property by virtue of which mass persistently retains its existing velocity unaltered, so long as it is not acted on by external force.

The nature of force is such that it needs time in which to change the velocity of mass.

Mass is inert, in the sense that it cannot of itself make any change in its motion.

External force acting on mass produces in it acceleration only; it does not produce any definite velocity unless it acts continuously for some definite interval of time.

Example. When it is observed that a given mass M has moved in a given interval from a point A to another B, we can from this infer nothing as to whether an external force has acted on the mass during that interval or not. The mass M may have moved over that distance by reason of its inertia; by virtue of which it persisted in moving from A to B with uniform velocity.

But when it is observed that a mass M has at one instant a certain velocity and that after some interval it has a different velocity, we are able to state that, for some part at least of that interval, it has been acted on by some external force.

50. When two equal forces act simultaneously throughout some interval each on one of two masses M and M', they produce in each mass the same number of pound-velos.

The forces are not necessarily uniform, provided they are always equal; and when the masses M and M' are different the velocities produced will be different.

51. It happens that the action of two exactly equal forces, acting simultaneously for the same interval of time on different masses is a phenomenon which is constantly taking place.

52. No external force can be applied to a mass without the mass exerting an equal and opposite force on the body which applies the force. This is Newton's Third Law. 'To every action there corresponds an equal and opposite reaction.' [See Chapter XI.]

53. DEF. A stress is that which consists of two equal and opposite forces together forming an action and its reaction.

Examples. The pressure of a heavy mass on the ground on which it rests, and the pressure of the ground on the mass, together form a stress.

The tension of a string at any point of its length is a stress.

54. When two particles act and react the one on the other, the stress causes no alteration in the total massvelocity of the two particles; that is, the stress has no effect on the algebraical sum of their mass-velocities.

For a stress consists of two equal and opposite forces acting simultaneously; so that whatever amount of massvelocity the action produces in one mass in any interval, the reaction produces in the other mass in the same interval an exactly equal amount in the opposite direction.

Example i. Let two particles A and B of m lbs. and m' lbs. be connected by a light string. The tension of this string is a stress ; so that whatever force is applied by the string and for whatever interval to the particle A, an exactly opposite and equal force is applied for the same interval by the string to the other mass B. So that, if mv poundvelos is the additional mass-velocity produced in the course of any interval by the tension in A, and m'v pound-velos the additional massvelocity produced in the same interval by the tension in B, then mv and m'u are equal and opposite; so that, m'o'= - mv;

mv + m'v=o. Suppose for example that the masses always have the same velocity,

v=v, then

(m + m') v=o; therefore v=0. Hence it appears that an internal stress between two particles, which cannot move relatively to each other, has no effect on their velocity,

or,

so that

m + m

Example ii. Two masses m lbs. and m' lbs. placed on a smooth horizontal plane are connected by a light inextensible string; a force of f poundals is applied to the mass m in the direction of the string, which is tight; find the acceleration produced; also the tension of the string.

1. The masses are prevented by the string from moving relatively to each other; hence, we may consider the tension of the string as an internal stress, and consequently the two masses as a single mass.

Thus, we have a mass of (m + m') lbs. acted on by f poundals; therefore the acceleration a is given by the equation f=(m + m') a;

f that is, the acceleration is

celos

(i). II. To find the tension of the string we must consider each component of the stress separately.

Let the string apply an external force T poundals to the mass m; it will consequently apply an external force T poundals to the mass m'.

Then, if a celos be the acceleration (which is the same for each mass, for the string is tight and inextensible) we have, f-T= ma ;

[from the motion of m and,

T= m'a;

[from the motion of mi whence, by addition, f=(m + m') a

(i),

m' wherefore

ne + m'

m' The tension required is

f poundals

(ii).

T=

if.

m + m'

EXAMPLES. XVI. 1. An engine of 40 tons is attached to a train of 6 carriages of 10 tons each; the train is moving with 5 celos; neglecting friction and the rotatory motion of the wheels, find the tension of the coupling between the engine and the next carriage.

2. In the train in Question 1 find the tension of the coupling between each pair of carriages.

3. In the train in Question 1, what force is the engine exerting?

4. The train in Question 1 moving at the rate of 30 miles an hour is stopped by brakes on the engine in a quarter of a mile; find

(i) the force exerted by the brake,
(ii) the stress between the engine and the next carriage,
(iii) the stress between the two last carriages,

(iv) if the brakes on the engine exerted the same force in all cases, how far would the engine alone, going 30 miles an hour when the brakes are applied, run before coming to rest ?

5. An engine which exerts a horizontal force sufficient to generate an acceleration a celos in its own mass m lbs., is attached to 3 carriages whose masses are £m lbs., £m lbs. and km lbs. respectively; find the acceleration of the train and the stress between each carriage, neglecting friction and the rotary motion of the wheels.

6. An engine which exerts a horizontal force p poundals, sufficient to generate in its own mass; a celos, is attached to n carriages each of which is Ath of the mass of the engine; find the acceleration of the train and the stress between the 4th and 5th carriages, neglecting friction etc.

55. The laws of motion may be illustrated and tested experimentally by the aid of Atwood's machine.

56. Atwood's machine may be described as follows.

A wheel or pulley, made as light and as free from friction as possible, is fixed at a convenient height from the ground; over the pulley is passed a light inextensible string, usually made of silk; from the ends of the string are suspended two masses M and M'. Each mass is acted on only by its own weight and by the tension of the string.

I. It will be found that we can arrange the masses so that they move with a very small acceleration compared with that produced in a mass falling freely under the action of its own weight; so that we can ascertain by actual measurement what kind of motion is produced by force in mass; we can thus test the statement that force produces in mass that kind of motion which has been defined as uniformly increasing velocity.

II. We can test by observation the statement that mass when acted on by no resultant external force moves with uniform velocity.

III. Since the mass of the string is very small, it causes no appreciable effect on the tension; if we also neglect the effects produced by the mass of the wheel and the friction of the axle (which are both comparatively small), then the tension remains unchanged throughout the whole length of

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