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Problem 714.
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An ideal spring of negligible mass is 12 centimeters long when nothing is attached to it.
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That's this right here, this length.
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When you hang a 3 .15 kilogram weight from it, you measure its length to be 13 .4 centimeters.
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And that's going to be this length right here with its dotted box showing the next position when it's stretched.
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If you wanted to store 10 joules of potential energy in the spring, it will be its total length.
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Assume it continues to obey hooks law okay so hooksaw here is going to be the basic outline of what we're going to be doing we need to calculate how much energy stored in this and in this spring and that and for that we need to know what the spring constant is because it is in the equation for the potential energy of a spring so we need to find what that is since we're not given it so that's going to be what we need to find and in order to do that we're given some information we're told the unstretched and the stretch length is for a certain mass so what we can do is we can use the two different positions of this mass when it's when it gives an unstretched and stretched spring to calculate that spring constant so the what we're going to do is we're going to set up conservation of energy so we have initially u elastic of the spring plus you for the potential energy of gravity and these are both in position one all right and that's going to equal to the potential energy stored in the spring at position two plus the potential from gravity at position two okay so in order to cancel some of this stuff out we know initially that the spring is unstretched so there's no energy stored in it initially so that means that this term is going to go away.
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And also we can define the position when the gravity is at zero to be at this point here.
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So we can set that to have y equals zero for gravity.
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So that means that we're going to cancel out this term here and we're going to have our gravity in terms of this length up from this point.
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So that means that what we're going to have is we're going to write the expression for each of these two terms here.
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So we're going to have m .g.
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And we can write some height there for now.
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And then we're going to have that equal to expression for the elastic potential energy of the spring.
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And that's going to be one half k x squared.
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And now x here is going to be the change in the spring's length here.
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So what we need to do is we need to have this point right here.
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You need a point where x is equal to zero because this string is unstretched there and it goes down to this point here.
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So that means that the amount of displacement of x, this delta x, is going to be a difference between those two positions.
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So that delta x is just going to be 13 .4 minus 12, 1 .1 .1 .4 .1 .1...