WEBVTT
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This is Giancoli Answers
with Mr. Dychko.
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We have a mass hanging off
the end of this string here.
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And the mass experiences
a weight downwards
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equal to mg
and a tension force upwards.
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This tension force is going to be
the tension in the string
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and it's going to be equal
to the weight of this hanging mass
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because the mass is not moving.
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And the string has a length of l
that we don't really know what it is,
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but we know that it can vary
from 10 centimeters up to 1.5 meters,
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and our job is to figure out
how many standing wave patterns
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can fit in between those lengths.
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We have an oscillator here
moving at 60 hertz
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and it's moving with a small amplitude
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and that's important because we can
still consider this end to be a node.
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Nodes don't move at all.
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And now since the oscillator
is moving just a little bit,
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we can still think of it as a node.
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And let's figure out
what the wavelength of this...
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The wave in the string will be.
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And then we'll consider this to be a node
and this to be a node.
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And we'll have
these standing wave patterns
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that are possible here.
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This one is the shortest
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distance from the pulley
to the oscillator possible,
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and that contains half
of a wavelength.
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And there's a node here
and a node there.
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And then the next
smallest possibility would be
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with this node at the oscillator
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and then the next node
at the pulley being here
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one full wavelength
away from the oscillator.
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And then we can just guess
and check and keep trying
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the next standing wave pattern here
with a node at the weight or pulley
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and the node at the oscillator here,
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and that's a wavelength
plus half a wavelength.
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So, that's 3 over 2 wavelengths.
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And then we'll try this one too
and find that it doesn't fit.
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So, what is the wavelength
first of all?
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Well, we know wave speed is going
to be frequency times wavelength.
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And then we can divide
both sides by f
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and we get wavelength is wave speed
divided by frequency.
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And wave speed is
the square root of tension
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divided by mass per unit length.
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And the tension we already
figured out is mg.
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So, we'll substitute for that here
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and I just put
this square root μ in the bottom.
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And so we have wavelength
the square root mg
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divided by frequency
times square root mass per unit length.
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And so that's a square root
of 0.08 kilograms
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times 920 newtons per kilogram
divided by 60 hertz
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times 3.5 times 10 to the minus 4
kilograms per meter square rooted
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and that gives 0.7888 meters
is the wavelength.
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So, this distance here, l,
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in the first case will be
wavelength divided by 2
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and that's going to be 0.394 meters.
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And so that works because
it's greater than 10 centimeters
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and less than 1.5 meters.
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And then the next possibility is
with this node at the pulley
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and this node at the oscillator.
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And that's two antinodes
or one full wavelength here.
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And that's 0.789
and that works.
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And then the next pattern
that's possible is this pattern
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with l being 3 halves
you could say,
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here's one half wavelength
here's another half wavelength,
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one half, two halves, three halves.
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And that's 1.18 meters
and that works.
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And then this pattern here
which is two full wavelengths
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does not work because it's 1.578 meters
and that exceeds 1.5
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and so three standing
wave patterns are possible.