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$\bullet$ Three identical 6.40 $\mathrm{kg}$ masses are hung by three identicalsprings, as shown in Figure $5.59 .$ Each spring has a forceconstant of 7.80 $\mathrm{kN} / \mathrm{m}$ and was 12.0 $\mathrm{cm}$long before any masses were attached to it.(a) Make a free-body diagram of each mass.(b) How long is each spring when hanging asshown? (Hint: First isolate only the bottommass. Then treat the bottom two masses as asystem. Finally, treat all three masses as asystem.)

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a) See $\mathrm{F.B.D.}$b) $L_{1}=12.8 \times 10^{-2} \mathrm{m}, L_{2}=13.6 \times 10^{-2} \mathrm{m}, L_{3}=14.4 \times 10^{-2} \mathrm{m}$

Physics 101 Mechanics

Chapter 5

Applications of Newton's Law

Motion Along a Straight Line

Motion in 2d or 3d

Newton's Laws of Motion

Applying Newton's Laws

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right. So here's the situation. You have different masses hanging from different springs. So massive 12 and three year indicated here paying from strings Springs 12 and three is indicated here. Okay, so the free body diagrams for these for these different cases will be follow. So for spring one, one and two will be similar for spring one. You have you have a case of X one, the stretching for sacking upwards. And to force that downwards. One of them is a force of gravity. And she which, by the way, is the same for all three masses. And you also have a case of X two, which is the stretching force from the second spring. Um, acting in the same direction is gravity. Not for the okay. And for the suffer. The second second mass. You have a very similar situation. Case of X one is at heart. He's a next to the second upwards of stretching for us on DH. You have gravity acting downwards. Say them to, and then you have a case of extreme stretching for us from the third spring acting. Uh, now, damn it. And for the third mass, you have these two forces case of X three is just stretching for a second upwards and that is bounced by gravity acting downwards. Okay, because there's no strings attached spring attached below that so well the easiest way to go about doing doing part B is by by looking at you system separately. So let's start it sitting three and here f equals And so we resolve forces after in the white direction will be called zero And so this just means that K X tree will be called energy So x three is mg over care and we know Mr Beene on ji is six point lips Jia's 6.4 kilograms tons 9.8 meters per second squared over care which is seven thousand 7.8 killing you So one killing you is 1,000 So 7,800 per meters on DSO This is just eight times 10 to the negative three meters attended many of two meters is one centimeter So this is 10.8 centimeters centimetres. Okay, now let's look at the second. He was a different uh I think it's look it the second mass this case the Why don't why the net. The wind direction force is net. Force is equal Toe cakes too. Ah, minus K x three minus energy called zero. So kicks too equals cakes too. Plus Mt. Right. And so we already found the cakes. To me is mg from here, So okay, X two is just two mg as a result. Therefore, X two is just two times and she over Kay. And so this is just twice of what we have found for X tree, seriously, 1.6 times to the negative two meters or 1.6 centimeters and and then for part and then for the first mass. Very similarly, you have kicks one as you call to K X two plus mg and and so kekst too. We just found to be two mg. And so this is just three mg, implying that X one is three mg over care and we already know that mg over K is 0.8 centimetres. So this would be three times that that we have three times 30.8 centimeters. That's 2.4 centimeters. Okay, um and finally, we want the lengths off the entire strings. So the length of the entire string let's use a different color Length of the entire string will be for any for each string. So for l one, it will be l not plus X one for L too. It will be well, not the UN stretch length sex too. And similarly, lt is all not possess X three. Um, we have we know that l not is 12 centimeters. Therefore, L one is 12 plus 2.4 s. Oh, it's 14.4 centimeters. Two is 12 plus one point to 1.6. So that's 13.6 centimeters and three is 12 plus 0.8. So that's 12.8 centimeters and that's it.

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