Some Mechanical Questions

Excerpts

• Some Mechanical Questions

  Hence the forces that act on the machine and are brought into play by the various parts may be altered in different proportions, and thus the machine may be incapable of producing results similar to those which can be produced by the model.

• Some Mechanical Questions

  But as the velocity of the boat increases, a time will arrive when the pressure of the wind is only just able to balance the resisting force which is caused by the sail moving through the air.

• Some Mechanical Questions

  Hence the usual movements of the crew in the boat do not sensibly move the centre of gravity of themselves and the boat, but this does not apply to an impulsive movement, and if the crew in making a jerk move their centre of gravity towards the bow $n$ times more rapidly than it returns after the jerk, then the boat is impelled forwards at least $n$ times more than backwards: hence on the whole the motion is forwards

• Some Mechanical Questions

  Thus, if a number of dominoes or draughts are arranged in a vertical pile, a sharp horizontal blow on one of those near the bottom will send it out of the pile, and those above will merely drop down to take its place---in fact they have not time to change their relative positions before there is sufficient space for them to drop vertically as if they were a solid body.

• Some Mechanical Questions

  In other words, if in order to cause a displacement work has to be done against the forces acting on the body, then for that displacement the equilibrium is stable, while if the forces do work the equilibrium is unstable.

• Some Mechanical Questions

  To establish this, Zeno argued that when Achilles had gone the $1000$ yards, the tortoise would still be $100$ yards in front of him; by the time he had covered these $100$ yards, it would still be $10$ yards in front of him; and so on for ever.

• Some Mechanical Questions

  So naturalists observe, a flea hath smaller fleas that on him prey.

• Some Mechanical Questions

  Montucla says that in his time it was not uncommon to see boxes of tin soldiers mounted on lead hemispheres, and when the lid of the box was taken off the whole regiment sprang to attention.

• Some Mechanical Questions

  Probably the meaning of the law is best expressed in Clifford’s phrase, that force is “the description of a certain kind of motion”---in other words it is not an entity but merely a convenient way of stating, without circumlocution, that a certain kind of motion is observed.

• Some Mechanical Questions

  A given agent in a given time can do only a definite amount of work. This is illustrated by the fact that although, by means of a rigid lever and a fixed fulcrum, any force however small may be caused to move any mass however large, yet what is gained in power is lost in speed---as the popular phrase runs.

• Some Mechanical Questions

  The assertion was that if Achilles ran ten times as fast as a tortoise, yet if the tortoise had (say) $1000$ yards start it could never be overtaken.

• Some Mechanical Questions

  Such a bottle is made of thin glass or varnished paper fixed to the plane surface of a solid hemisphere or smaller segment of a sphere.

• Some Mechanical Questions

  Of course it is only in isolated systems that the total amount of energy is constant, and, if a source of external energy can be obtained from which energy is continually introduced into the system, perpetual motion is, in a sense, possible; though even here materials would ultimately wear out.

• Some Mechanical Questions

  Montucla calculated the mass of the earth and, assuming that a man could work incessantly at the rate of $116$ foot-lbs. per second, which is a very high estimate, he found that it would take over three billion centuries, $3 \times 10^{14}$ years, before a mass equal to that of the earth was moved as much as one inch against gravity at the surface of the earth

• Some Mechanical Questions

  If all the parts of a model are magnified in the same proportion, say $m$, and if thereby a line in it is increased in the ratio $m:1$, then the areas and volumes in it will be increased respectively in the ratios $m^2:1$ and $m^3:1$.

• Some Mechanical Questions

  For example, if the side of a cube is doubled then a face of it will be increased in the ratio $4: 1$ and its volume will be increased in the ratio $8:1$.

• Some Mechanical Questions

  Again, if the linear dimensions of a man of height $5$ ft. $10$ in. were increased by one-seventh his height would become $6$ ft. $8$ in., but his weight would be increased in the ratio $512: 343$ ( about half as much again), while the cross sections of his legs, which would have to bear this weight, would be increased only in the ratio $64:49$; thus in some respects he would be less efficient than before.

• Some Mechanical Questions

  Hence, if the wind makes the same angle $\alpha$ abaft the beam that the sail makes with the keel, the velocity of the boat will be greater than the velocity of the wind.

• Some Mechanical Questions

  The chief cause for this result seems to be that the friction between the boat and the water retards all relative motion, but is not great enough to materially affect motion caused by a sufficiently big impulse.

• Some Mechanical Questions

  Thus, if a current of air is moving in a tube, the pressure on the sides of the tube is less than when the air is at rest---and the quicker the air moves the smaller is the pressure.

• Some Mechanical Questions

  If anyone blows steadily through the tube so formed, the paper will be sucked in instead of being blown out.

Equations

• Some Mechanical Questions

  a + a/n + a/n^2 + a/n^3 + \dotsb

  The time for a point moving uniformly along the equiangular spiral to reach the pole is the infinite sum of the times for successive convolutions, each 1/n of the one before.

• Some Mechanical Questions

  an/(n-1)

  The closed-form value of the infinite geometric sum, so the point reaches the pole in a finite time an/(n-1) seconds despite circling it infinitely many times.

• Some Mechanical Questions

  W\!AR = \theta

  The angle between the wind direction WA and the keel AR is called theta.

• Some Mechanical Questions

  BAS = \alpha

  The angle between the keel AB and the sail AS is called alpha.

• Some Mechanical Questions

  W\!AL=\theta + \alpha

  The angle between the wind direction and the sail AL equals theta plus alpha.

• Some Mechanical Questions

  v \sin \alpha = u \sin (\theta + \alpha)

  For steady motion of the boat, the boat speed times the sine of the sail angle equals the wind speed times the sine of the sum of the wind and sail angles, so that the resultant pressure normal to the sail vanishes.

• Some Mechanical Questions

  v>u

  The boat moves faster than the wind when sin(theta + alpha) is greater than sin alpha.

• Some Mechanical Questions

  \sin (\theta + \alpha)>\allowbreak\sin \alpha

  If sin(theta + alpha) is greater than sin alpha, then the boat's speed exceeds the wind speed.

• Some Mechanical Questions

  \theta + \alpha = \frac{1}{2}\pi

  With the sail angle fixed, the boat speed is a maximum when theta plus alpha is a right angle, i.e. theta is the complement of alpha.

• Some Mechanical Questions

  v = u \cosec \alpha

  At the maximum, the boat speed equals the wind speed times the cosecant of the sail angle, which is greater than the wind speed.

• Some Mechanical Questions

  v \sin \alpha = u \sin \phi

  For a boat running close to the wind, the steady-motion condition is written with phi, the angle WAS between the wind and the sail.

• Some Mechanical Questions

  \phi = \text{angle } W\!AS = \pi - \theta - \alpha

  The angle phi between the wind and the sail is pi minus theta minus alpha.

• Some Mechanical Questions

  v = u \sin \phi \cosec \alpha

  For a boat running close to the wind, the boat speed is the wind speed times sin phi times the cosecant of alpha.

• Some Mechanical Questions

  w =\allowbreak v \cos BAW =\allowbreak v \cos (\alpha + \phi) =\allowbreak u \sin \phi \cosec \alpha \cos (\alpha + \phi)

  The component velocity of the boat in the teeth of the wind equals the boat speed times the cosine of the angle BAW, which is alpha plus phi, giving w in terms of u, phi and alpha.

• Some Mechanical Questions

  \phi = \frac{1}{4}\pi - \frac{1}{2}\alpha

  With alpha constant, the component velocity w in the teeth of the wind is a maximum when phi equals a quarter of pi minus half of alpha.

• Some Mechanical Questions

  w = \frac{1}{2}u (\cosec\alpha - 1)

  At the maximum, the component velocity of the boat in the teeth of the wind is half the wind speed times (cosecant alpha minus 1).

• Some Mechanical Questions

  \sin\alpha< \frac{1}{3}

  The component velocity in the teeth of the wind is greater than the wind speed when sin alpha is less than one third.

• Some Mechanical Questions

  p = \Pi\alpha^{-v^2}

  In an elastic perfect fluid whose pressure is proportional to its density, the pressure falls exponentially with the square of the steady velocity of the fluid.

Problems

No exercises in this chapter.