hello and welcome to the chemistry
solution this tutorial is on dimensional
analysis and this tutorial is part two
in a series where we'll be covering some
more complicated dimensional analysis
problems if the standard value of
Earth's gravitational acceleration at
sea level is 9.81 m/s^2
what is the value in kilometers per hour
squared in this problem we not only have
to change one set of units but we need
to change two sets of units we need to
change meters to kilometers and we need
to change second squared to hour squared
we'll also be looking at how to convert
a unit raised to a power
in this example second squared two hours
squared let's start by writing the
information that we're given 9.81 meters
on the top per second squared on the
bottom
we'll need a conversion factor to
convert from meters to kilometers
looking at the conversion factors we
have available to us we know that 1,000
meters is equal to one kilometer and
because we have meters on the top and
our problem and we want meters to cancel
we'll need to put the value with the
units of meters on the bottom of our
conversion factor this means that one
kilometer must go on the top and this
allows us to cancel out the units of
meters so if we were to stop our problem
now we would have an answer in
kilometers per second squared now we
need to convert second squared to our
squared and we'll look at how to convert
a unit that's raised to a power we know
that we'll need to use a conversion
factor and we can see that 60 seconds is
equal to one minute because we want
seconds on the top we know that minutes
must go on the bottom now what you'll
see is that we have units of seconds on
the top and units of second squared on
the bottom this allows us to cancel out
only one of the units of seconds on the
bottom remember that second squared is
just equal to seconds times seconds
so after we cancel out one units of
seconds we're still left with units of
seconds on the bottom so to convert from
second squared to minute squared we'll
need to apply this conversion factor
again having 60 seconds on the top again
allows us to cancel out the other unit
of seconds on the bottom so now we're
left with kilometers per minute times
minute or kilometers per minute squared
now I think the best way to solve these
dimensional analysis problems is always
to write out your conversion factor as
many times as you need to to cancel out
all the units on the bottom so because
we had second squared or seconds times
seconds on the bottom we need two
conversion factors with seconds on the
top in order to cancel those out but if
you'd like you can also just square the
unit's on your conversion factor as well
as squaring the number and write your
conversion factor like this here we have
60 squared second squared per 1 squared
minute squared and what you'll see we
did here is we just took the conversion
factor 60 seconds is equal to 1 minute
and squared the entire conversion factor
that's the same thing we did before this
is just a more concise way of writing
the same thing and you'll see now that
second squared on the top is able to
cancel a second squared on the bottom
we're going to do the same thing for
this next conversion factor because we
know that 60 minutes is equal to one
hour
we know that the minutes will have to go
on the top and hours will go on the
bottom and here instead of writing out
the conversion factor twice I squared
the whole thing so remember if you're
going to square the unit so you also
have to square the number that goes
along with those units and so here
because we squared minutes we also need
to square 60 and because we squared
minutes and 60 we also need to square
hours and square one this allows us to
cancel the minute squared on the top
with the minutes squared on the bottom
and gives us an answer of kilometers per
hour squared and when you punch this
into your calculator you'll come up with
an
answer of 1 point 2 7 times 10 to the
5th km/h squared let's try one more
example like this one nanogram per
nanometer cubed equals how many
kilograms per meter cubed we'll start by
writing the information that's been
given one nanogram per nanometer cubed
again we know we need to use a
conversion factor and this time let's
convert nanometers cubed to meters cubed
first remember whenever you have a
problem where you're converting two
different sets of units the order in
which you convert those units is
irrelevant now remember a nanometer is a
very very small unit of measure there
are 10 to the 9th nanometers in every
one meter or likewise sometimes you'll
see this conversion factor written 10 to
the negative ninth meters is equal to
one nanometer these conversion factors
are equal so it doesn't matter which one
you use I'm going to go ahead and use 10
to the 9th nanometers for every one
meter now looking at our units we have
nanometers cubed on the bottom and
nanometers on the top this allows us to
only cancel one units of nanometers when
we write this conversion factor again
now we have nanometers cubed on the
bottom and nanometers times nanometers
or nanometers squared on the top so now
we're able to cancel two units of
nanometers we need to write this
conversion factor one more time so we
have nanometers times nanometers times
nanometers on the top and nanometers
cubed which is nanometers times
nanometers times nanometers on the
bottom writing the conversion factor
three times allows us to cancel
nanometers cubed on the bottom with
nanometers times nanometers times
nanometers or nanometers cubed on the
top you'll see that by writing the
conversion factor three times will also
now be multiplying our initial value
times 10 to the ninth three times and
this is important to do in order to
arrive at the correct answer if we were
to stop this problem now we would have
units of nanograms
our meters times meters times meters or
meters cubed so we have successfully
converted nanometers cubed to meters
cubed now we need to convert nanograms
to kilograms because the prefix nano
means the same thing regardless of
whether it's in front of grams or meters
you'll also see that we can use a
similar conversion factor 10 to the 9th
nanograms is equal to one gram and
because we want nanograms on the top to
cancel will write nanograms on the
bottom of our conversion factor this
allows us to cancel out units of
nanograms and if we were to stop this
problem now we would have units of grams
per meter cubed to convert to kilograms
we'll use the conversion factor 10 to
the third grams for every one kilogram
and we'll put 10 to the third grams on
the bottom in order to cancel out units
of grams and kilograms on the top this
allows us to cancel out units of grams
and now we're left with the final answer
in units of kilograms per meter cubed
and when you punch these numbers into
your calculator you'll come up with 1
times 10 to the 15th kilograms per meter
cute thanks for watching the chemistry
solution we hope you enjoyed this
tutorial
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