WEBVTT

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This is Giancoli Answers
with Mr. Dychko.

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We know the total energy is 1/2
times spring constant

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times amplitude squared.

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And we can solve this for <i>k</i>

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and say that it's 2 <i>E</i>
over <i>A</i> squared

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multiplying both sides by 2
and dividing both sides by <i>A</i> squared

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and then switching the sides around.

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And we get <i>k</i> is 2<i>E</i> over <i>A</i> squared.

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We also know that when the spring
is fully compressed,

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the net force
that the block experiences

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will be <i>ma</i>
because that's Newton’s second law.

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But we also know that the net force
will also be the spring force

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because that's the only horizontal force
that will be acting

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after the, you know,
initially compressing force is removed,

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the only force
that exists is the spring pushing back.

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And so <i>ma</i> equals <i>k</i> times

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the amount of the spring is compressed

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which is going to be
equal to the amplitude

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since it's compressed maximally here.

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And so we solve for <i>k</i>
by dividing both sides by amplitude.

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And <i>k</i> is mass and acceleration
divided by amplitude.

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So, each of these things equals <i>k</i>,

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so that means
they're equal to each other.

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So, <i>ma</i> over amplitude equals 2

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times total energy
divided by amplitude squared.

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And we can solve for <i>m</i>
and multiply both sides by <i>A</i>

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and divide both sides by acceleration.

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And one of these <i>A's</i> cancels
on the bottom

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and you have 2 <i>E</i> over amplitude
times acceleration.

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So, I'm answering part B first
as it turns out

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just the way my algebra worked.

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So, we have 2 times 3.6 joules
divided by 0.13 meters amplitude

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times 12 meters
per second squared acceleration

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which gives 4.6 kilograms.

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And then we have an equation
for spring constant,

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mass times acceleration
over amplitude.

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And now that we know mass
we can substitute into that,

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4.6154 kilograms
times 12 meters per second squared

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divided by 0.13 meters

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gives about 430 newtons per meter
for the spring constant.