### Video Transcript

For a satellite to follow a
circular orbit around Earth at a radius of 10,000 kilometers, what orbital speed
must it have? Use a value of 5.97 times 10 to
24th kilograms for the mass of Earth and 6.67 times 10 to the negative 11th meters
cubed per kilogram per second squared for the value of the universal gravitational
constant. Give your answer to three
significant figures.

Okay, so here’s the Earth with a
mass of 5.97 times 10 to 24th kilograms. And here is the circular orbit with
a radius of 10,000 kilometers, as measured from the center of the Earth. Finally, here’s a satellite moving
along the orbit with an unknown speed. For the satellite to maintain a
constant circular orbit, it must be experiencing a centripetal force which comes
from the force of gravity that the Earth exerts on the satellite.

Let’s call the mass of the
satellite lowercase 𝑚, which is not a quantity that we’re given in the problem. Using lowercase 𝑚, 𝑣, 𝑟, capital
𝑀, and the universal gravitational constant. We can write 𝐹 c, the centripetal
force, is equal to the mass of the satellite times the square of the orbital speed
divided by the orbital radius. Also, 𝐹 g, the force of gravity
that Earth exerts on the satellite, is equal to the universal gravitational constant
given the symbol capital 𝐺 times the mass of Earth times the mass of the satellite
divided by the square of the orbital radius.

Since the centripetal force is
provided by the gravitational force, we can equate these two quantities and then
solve for the orbital speed. We get 𝑣 is equal to the square
root of 𝐺 times capital 𝑀 divided by 𝑟, whereas we can see the mass of the
satellite does not appear in this final expression. In our question, we’re given a
value for the universal gravitational constant. So combining that with our known
values for orbital radius and mass of Earth, we should be able to plug in to get a
value for the orbital speed.

When we actually put in those
numbers, we find that 𝑣 is equal to the square root of 6.67 times 10 to the
negative 11th meters cubed per kilogram per second squared times 5.97 times 10 to
the 24th kilograms divided by 10,000 kilometers. Let’s start with two
simplifications involving the units. We have a factor of per kilograms
and a factor of kilograms. And per kilograms times kilograms
is just one.

Second, we’ll need to convert
kilometers to meters in the denominator to match the meters already present in the
numerator. Recall that one kilometer is by
definition 1000 meters. So, 10,000 kilometers is 10,000
times 1000 meters or, converting to scientific notation, 10 to the seventh
meters. Finally, meters cubed in the
numerator divided by meters in the denominator is meters squared. Let’s rewrite this expression
separating the numbers, the powers of 10, and the units.

Written out like this, using the
commutativity and associativity of multiplication, we can now calculate the value of
each of these terms separately, take the square roots separately, and multiply all
those together to get the final result. Let’s start with the units. Meters squared per second squared
is meters per second squared. Moving on to the powers of 10, our
final result will have a base of 10. And to get the exponent, we add the
exponents in the numerator and subtract the exponents in the denominator. Negative 11 plus 24 is 13 minus
seven is six. So, that whole term reduces to 10
to the sixth.

Finally, 6.67 times 5.97 is equal
to 39.8199. To finish the calculation, we
recall that the square root of a product of several terms is equal to the product of
the square root of those terms. So, we have that 𝑣 is equal to the
square root of 39.8199 times the square of 10 to the sixth times the square root of
meters per second squared. Okay, so let’s work out these
square roots.

To take the square root of a
squared quantity, well, that’s just the quantity itself. So, the square root of meters per
second squared is just meters per second. This is a good intermediate result
because meters per second is a unit of speed. So, we see that we’re looking for a
speed, and our final answer will have units of speed. In general, to take the square root
of any quantity raised to a power, simply halve the power. So, the square root of 10 to the
sixth is 10 to the third.

Lastly, we have the square root of
39.8199. For this, we just need a
calculator. The first several digits of that
result are 6.3103 et cetera. Now, we have our answer as a number
times the power of 10 times some units, which is a very useful form for expressing
it to three significant figures. To express our answer this way, we
simply need to find the three significant figures of the number portion and then
carry the power of 10 and the units to the final answer.

To identify some number of
significant figures, we count that many digits, starting from the first nonzero
digit and going left to right. For our number, the first
significant figure is six and the second is three. To find the third significant
figure, since that’s the last one that we’re looking for, we have to round. So, we look at the fourth digit,
which is zero. And since zero is less than five,
one rounds to one. So, our number to three significant
figures is 6.31.

Now, let’s just finish out the
multiplication. 6.31 times 10 to the third is 6,310
times meters per second gives us a final answer of 6,310 meters per second to three
significant figures. And this is the orbital speed that
a satellite would need to maintain to have a circular orbit around Earth with a
radius of 10,000 kilometers.