A jackhammer (or pneumatic hammer) uses compressed air to drive a metal piston up and down inside a cylinder. Each time the piston nears the top of the cylinder, it opens a valve that allows compressed air to flow above it and push it downward. Each time the piston reaches the bottom of the cylinder, it opens a valve that allows compressed air to flow below it and push it upward. Thus the compressed air makes the piston shuttle up and down very rapidly.
But while the piston rebounds gently from a cushion of air at the top of the cylinder, it collides suddenly with a metal bar at the bottom of the cylinder. That metal bar is the top end of the drill bit that the jackhammer uses to cut into pavement. Each time the piston moves downward, it pounds the drill bit a little farther into the pavement. The enormous force that pushes the bit into cement comes from the enormous force needed to stop the descending piston and to accelerate it upward. The drill bit pushes up on the piston very hard and the piston pushes down on the drill bit very hard. These two forces are equal and opposite, as they must be (Newton's third law of motion.) The piston ends up moving upward and the drill bit ends up moving downward.
Yes. If there were 5 segments in the multiple pulley, then you would have to pull down on the bottom of the multiple pulley with a force that was 5 times the magnitude of the object's weight in order to lift the object at constant velocity. But the object would also rise 5 times as fast as the end of the multiple pulley would descend.
When the string you pull on comes down from the top pulley, it doesn't exert its tension on the thing being lifted so it doesn't count when add up the strings. But when the string you pull on comes up from the bottom pulley, that string is also helping to lift the object. That string does count. Thus if the multiple pulley has 5 segments going up and down between the two pulleys and one more segment going up to your hand, the total number of segments lifting the object is 6 and that object experiences an upward force equal to 6 times the tension in the string.
As you have noticed, buses, trucks, and trains often use air as the hydraulic fluid in their braking systems. That's because air is cheap and non-toxic, so that spilling it isn't a problem. While air's compressibility makes it a bit more complicated to work with than a liquid hydraulic fluid, it still works well in power braking systems.
When you pull on the string with a 10 N force, you create 10 N of tension in that string. If there is less tension anywhere in the string, then that portion of the string will accelerate toward the side with more tension. That's why the tension in each string of a multiple pulley is 10 N when you pull on its loose end with a force of 10 N. The 5 strings are really just parts of the same string and that string has to have 10 N of tension in it.
I'm afraid that your friend is right—liquids are slightly compressible. A compressible material is one that experiences a decrease in volume when it's exposed to an increase in pressure. Gases are highly compressible—they change volume dramatically with changes in pressure. Liquids are said to be incompressible—they change volume very little with changes in pressure. But very little isn't zero. A liquid is essentially incompressible because its atoms and molecules are touching one another and, since those atoms and molecules have relatively fixed sizes, it's hard to pack them closer together than they already are. But increases in pressure do cause those atoms and molecules to move slightly closer together and the liquid does becomes denser and occupies less volume. The effect is small enough that it has almost no effect on most hydraulic systems—the pressurized fluid loses only parts per million of its volume as you squeeze it with normal pressures. All you really care about in a hydraulic system is that over the range of pressures used, the fluid involved doesn't change volumes much. Thus if you keep the pressure changes small enough, even air can be used in a hydraulic system. For example, pneumatic tube delivery systems are essentially air-operated hydraulic systems. But if the pressure changes are large enough, even liquids and solids can be highly compressible. In fact, plutonium-based nuclear weapons use high explosives to crush spheres of solid plutonium, already one of the densest materials in existence, to several times solid density. You wouldn't think of plutonium as compressible, but under these astronomical pressures it compresses almost like a gas.
Your solution should work nicely—the pulley and weight system should protect your cable from breaking because the weights should maintain a constant tension in the line. As the tree swings back and forth, the weights should rise and fall while the tension in the cord remains almost steady. Obviously, if the rising weights reach the pulley the cord will pull taut and break, so you must leave enough hanging slack.
However, if the tree's motion is too violent, even this weight and pulley system may not save the cable. As long as everything moves slowly, the tension in the cord should be equal to the weight of the weights. But if the tree moves away from the house very suddenly, then the tension in the cord will increase suddenly because the cord must not only support the weights, it must accelerate them upward as well. Part of the cord's tension acts to overcome the weights' inertia. Just as a sudden yank on a paper towel will rip it free from the roll, so a sudden yank on your cable will rip it free from the weights. If sudden yanks of this type cause trouble for you, you can fix the problem by coupling the cord to the weights via a strong spring. On long timescales, the spring will have no effect on the tension in the cord—it will still be equal to the weight of the weights. But the spring will stretch or contract during sudden yanks on the cord and will prevent the tension in the cord from changing abruptly either up or down. The spring shouldn't be too stiff—the less stiff and the more it stretches while supporting the weights, the more effectively it will smooth out changes in tension.
As far as the weight of the weights, that depends on how much curvature you want in the cable supporting the feeders. The more weight you use, the less the cable will sag but the more stress it will experience. You can determine how much weight you need by pulling on the far end of the cable with your hands and judging how hard you must pull to get a satisfactory amount of sag.
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