This java applet is a simulation that demonstrates the motion of
oscillators coupled by springs. The oscillators (the "loads")
are arranged in
a line connected by springs to each other and to supports on the
left and right ends. The mass of each load and the stiffness (spring
constant) of
each spring can be adjusted.
At the top of the applet on the left you will see the string of
oscillators in motion. By default, the number of loads is set to 5.
To move the loads, click on one of them, drag it
slightly to one side and then release it.
Below the string you will see a graph showing each
normal mode's contribution to the motion.
There are two sets of terms; on top are the magnitude terms, which
shows the amplitude of each normal mode, and on
the bottom are the phase terms. Low-frequency modes are on the
left and high-frequency modes are on the right.
If you move the mouse over one of the modes, it will turn
yellow, and the motion of the corresponding mode will be
drawn underneath the line of oscillators in yellow (unless it's too small to see). So if you move the mouse
over all the modes, you can see each of the terms individually.
(One thing to keep in mind when looking at the magnitude of
each mode is that the scale is not linear.
If it were linear, it would look like the higher-frequency modes all had
zero magnitude because their contribution is so small.
For small magnitudes, the scale is logarithmic; then about 1/4 of the
way up the scale it switches to linear. The same is true for negative
magnitudes.)
You can modify the motion of the string in one of two ways. You can
click on it directly, or you can modify the normal modes.
The Setup popup allows you to view some predefined interesting
cases. The first two choices, 5 masses and 2 masses
are very simple. The next two demonstrate weak coupling; in both
cases, you will see that the load on
the right will oscillate for a while, and then stop; meanwhile the one on
the left will oscillate, and then will stop while the right one oscillates,
etc. This is because the two active modes
have frequencies that are very close, causing beats to occur. Another
way to look at it is that energy is slowly transfered from the left load
to the right and then back again.
The Mouse popup controls what happens when you click on the string.
The default setting is Pull string, which causes the load you click on
to be pulled to either side.
If you set the popup to Move load, you can edit the position
of a single load without disturbing the others (until you release
the mouse button).
If you set it to Modify masses,
you can modify the mass of one of the loads by clicking on it and
dragging the mouse up or down. The size of the load will be changed
to match its mass.
If you set it to Modify springs, you can modify the spring constant
of one of the springs by clicking and dragging up or down.
Springs with a high spring constant have a reddish color.
The Reset Positions button allows you to reset the positions
of all the loads to equilibrium.
The Reset Masses button allows you to reset the masses of
all the loads to
the default.
The Reset Springs button allows you to reset the springs to
the same spring constant.
The Stopped checkbox allows you to stop or start the simulation.
Occasionally two loads will collide, or one of the loads will hit the
edge. If this happens, the loads involved will turn red briefly,
and the normal modes will be changed to match the new motion of the
string. By default, the loads will collide inelastically, causing
them to move at the same velocity until they are pulled apart by
springs. The Elastic Collisions checkbox allows you to change
this so that they collide elastically.
When the Lissajous Figures checkbox is checked, and there are two
loads (or two
active modes), then the amplitudes of these modes (their normal coordinates)
will be plotted, producing a Lissajous figure. This makes the most sense
for cases of weak coupling.
The Simulation Speed slider controls how fast the simulation will
proceed.
The Damping slider controls how much damping there is. Damping is
a force that slows the string down. It is proportional to the speed of
the string, so high-frequency modes are damped more than lower ones.
The Number of Loads slider will adjust the number of loads on the
string. This can be set as low as one. If you reduce the number of
loads then you also reduce the number of normal modes.
Zip archive of this applet.
java@falstad.com
A far higher place must be assigned to Judaism among the competitors for the allegiance of Europe. The cosmopolitan importance at one time assumed by this religion has been considerably obscured, owing to the subsequent devolution of its part to Christianity. It is, however, by no means impossible that, but for the diversion created by the Gospel, and the disastrous consequences of their revolt against Rome, the Jews might have won the world to a purified form of their own monotheism. A few significant circumstances are recorded showing how much influence they had acquired, even in Rome, before the first preaching of Christianity. The first of these is to be found in Cicero’s defence of Flaccus. The latter was accused of appropriating part of the annual contributions sent to the temple at Jerusalem; and, in dealing with this charge, Cicero speaks of the Jews, who were naturally prejudiced against his client, as a powerful faction the hostility of which he is anxious not to provoke.330 Some twenty years later, a great advance has been made. Not only must the material interests of the Jews be respected, but a certain conformity to their religious prescriptions is considered a mark of good breeding, In one of his most amusing satires, Horace tells us how, being anxious to shake off a bore, he appeals for help to his friend Aristius Fuscus, and reminds him of217 some private business which they had to discuss together. Fuscus sees his object, and being mischievously determined to defeat it, answers: ‘Yes, I remember perfectly, but we must wait for some better opportunity; this is the thirtieth Sabbath, do you wish to insult the circumcised Jews?’ ‘I have no scruples on that point,‘ replies the impatient poet. ‘But I have,’ rejoins Fuscus,—‘a little weak-minded, one of the many, you know—excuse me, another time.‘331 Nor were the Jews content with the countenance thus freely accorded them. The same poet elsewhere intimates that whenever they found themselves in a majority, they took advantage of their superior strength to make proselytes by force.’332 And they pursued the good work to such purpose that a couple of generations later we find Seneca bitterly complaining that the vanquished had given laws to the victors, and that the customs of this abominable race were established over the whole earth.333 Evidence to the same effect is given by Philo Judaeus and Josephus, who inform us that the Jewish laws and customs were admired, imitated, and obeyed over the whole earth.334 Such assertions might be suspected of exaggeration, were they not, to a certain extent, confirmed by the references already quoted, to which others of the same kind may be added from later writers showing that it was a common practice among the Romans to abstain from work on the Sabbath, and even to celebrate it by praying, fasting, and lighting lamps, to visit the synagogues, to study the law of Moses, and to pay the yearly contribution of two drachmas to the temple at Jerusalem.335 Jeff’s hand was quietly coming down. “What happened to you?” begged Sandy. “Something new has come up, sir. I was waiting there by my ship a good while back, and I heard another one cruising and spiraling, shooting the field, I guess, because he came in and set down. My crate, just the way you ordered, was down by the grove, not in plain sight in the middle of the course. But Jeff set his ship down, left the engine running, and went off. I stayed hid to see what would happen, but when he didn’t come back, I thought I’d better go and find you—and see if it meant anything to you.” She stood alone, with the sticky, wet knife in her hand, catching her breath, coming out of the madness. Then she stooped, and pushing the branches aside felt about for her pistol. It lay at the root of a tree, and[Pg 80] when she had picked it up and put it back in the holster, there occurred to her for the first time the thought that the shot in the dead stillness must have roused the camp. And now she was sincerely frightened. If she were found here, it would be more than disagreeable for Landor. They must not find her. She started at a swift, long-limbed run, making a wide detour, to avoid the sentries, bending low, and flying silently among the bushes and across the shadowy sands. The year 1756 opened with menaces to England of the most serious nature. The imbecility of the Ministry was beginning to tell in the neglect of its colonies and its defences. France threatened to invade us, and a navy of fifty thousand men was suddenly voted, and an army of thirty-four thousand two hundred and sixty-three of native troops; but as these were not ready, it was agreed to bring over eight thousand Hessians and Hanoverians. To pay for all this it was necessary to grant excessive supplies, and lay on new duties and taxes. In presenting the money bills in the month of May, Speaker Onslow could not avoid remarking that there were two circumstances which tended to create alarm—foreign subsidies and foreign troops introduced, and nothing but their confidence in his Majesty could allay their fears, or give them confidence that their burdens would be soon reduced. There was, in fact, no chance for any such reduction, for wars, troubles, and disgraces were gathering around from various quarters. The first reverse came from the Mediterranean. MUCH to their amazement, the boys waked up the next morning in Nashville, and found that they had passed through the "dark and bloody ground" of Kentucky absolutely without adventure. After drawing and dividing the rations and cartridges. Si gave the boys the necessary instruction about having their things ready so that they could get them in the dark the next morning, and ordered them to disregard the bonfires and mirth-making, and lie down to get all the sleep they could, in preparation for the hard work of the next day. Then, like the rest of the experienced men, who saw that the campaign was at length really on, and this would be the last opportunity for an indefinite while to write, he sat down to write short letters to his mother and to Annabel. "Bully for the Wild Wanderers of the Wabash," Shorty joined in. "They're the boss regiment in the army o' the Cumberland, and the Army o' the Cumberland's the boss army on earth. Hooray for US Co. Le's have a speech. Where's Monty Scruggs?" "Bring a light, do—I can't abide this dark." Albert suddenly began to look uneasy. After all he was not really drunk, only a little fuddled. He walked straight, and his roll was natural to him, while though he was exceedingly cheerful, and often burst into song, his words were not jumbled, and he generally seemed to have a fair idea of what he was saying. "But I heard what the doctor said to you." "A purty accident—wud them stacks no more dry than a ditch. 'Twas a clear case of 'bustion—fireman said so to me; as wicked and tedious a bit o' wark as ever I met in my life." Calverley stept from the shadow of the cliff, and beheld a meteor in the sky, brightening and expanding, as the clouds opened, until it assumed the appearance of a brilliant star, of astonishing magnitude, encircled by dazzling rays, which, in a singular manner, were all inclined in one direction, and pointing to that part of the horizon where lay the rival of England—France. The foreman's face assumed a deeper hue than usual: he looked fiercely at the galleyman, but there was a determination in the weather-beaten face that made him pause ere he spoke. "Galleyman," he at length said, "you knew the business before you came: if you be so fond of saving old witches' lives, why didn't you say so, that I might not now be in this dilemma?" "No, no, not the boy," replied Merritt, rather impatiently.