Video Transcript
A one-penny coin has a mass of 3.56
grams. What is the magnitude of the
gravitational force between two one penny coins if the distance between the centers
of the two coins is 10 centimeters? Use a value of 6.67 times 10 to the
negative 11 meters cubed per kilogram second squared for the universal gravitational
constant. Give your answer in scientific
notation to two decimal places.
Let’s say that these are our two
one-penny coins. The distance between their centers
of mass, we’ll call it 𝑟, is 10 centimeters, and the mass of each coin, we’ll call
that 𝑚, is 3.56 grams. To solve for the gravitational
force between these masses, let’s recall Newton’s law of gravitation.
This law says that the
gravitational force between two masses, one with a mass 𝑚 one and the other with a
mass 𝑚 two, is equal to the product of those masses multiplied by what’s called the
universal gravitational constant, capital 𝐺, divided by the distance between the
centers of mass of these masses, 𝑟 squared. This equation tells us that we can
take any pair of masses anywhere in the universe and there will be a gravitational
force between them.
Therefore, according to Newton’s
law, there’s a gravitational force of attraction between our two coins. The magnitude of this force equals
the universal gravitational constant times the mass of the one coin 𝑚 multiplied by
the mass of the other coin, which is the same value 𝑚, divided by the square of the
distance between their centers of mass. We can write then 𝐹 sub g equals
capital 𝐺 times 𝑚 squared over 𝑟 squared.
Since we’re given a value to use
for the universal gravitational constant as well as values for the mass 𝑚 of each
coin and the distance 𝑟 between their centers of mass, we can substitute all three
of those values in to this expression. Notice that the universal
gravitational constant includes units of kilograms for mass. That’s the SI base unit of
mass. On the other hand, though, the
masses of our coins are given in units of grams. Similarly, 𝐺 uses distance units
of meters, the SI base unit of length, while in our denominator, our radius is given
in units of centimeters.
Before we can calculate 𝐹 sub g,
we’ll need to make these units of distance and mass agree. To do that, we’ll convert our
coin’s mass in grams to a value in kilograms and our distance in centimeters to a
distance in meters. One gram, we recall, is equal to 10
to the negative three or one one thousandth of a kilogram. And therefore, 3.56 grams equals
3.56 times 10 to the negative three kilograms. In a similar way, one centimeter
equals 10 to the negative two or one one hundredth of a meter. This means that 10 centimeters
equals 10 times 10 to the negative two meters.
The units in this expression now
all agree with one another. We can now calculate the force 𝐹
sub g. To two decimal places, it equals
8.45 times 10 to the negative 14 newtons. This force is much too small for us
to be able to detect, say, by holding these two coins a distance of 10 centimeters
apart. But nonetheless, this is the
magnitude of the gravitational force between the coins.
