### Video Transcript

Given that a planetβs mass and
diameter are three and six times those of Earth, respectively, calculate the ratio
between the acceleration due to gravity on that planet and that on Earth.

Recall that the acceleration due to
gravity π at the surface of a uniform sphere of mass π and radius π is equal to
πΊπ over π squared, where πΊ is the universal gravitational constant. This is the same formula as that of
a particle of mass π located at the geometric center of the sphere.

Letβs look at our two planets. Letβs say the Earth has a mass π
equals π naught and a radius π equals π naught. The mass of the other planet is
three times that of Earth. So its mass π is equal to three π
naught. And since its diameter is six times
that of Earth, its radius is also six times that of Earth. So π is equal to six π
naught.

There are a few ways in which we
could approach this problem. One way is to substitute the masses
of the Earth and the planet into the formula for acceleration due to gravity
explicitly and then find their ratio. Letβs do this first.

So, for Earth, the acceleration due
to gravity π E is equal to πΊπ naught over π naught squared. And for the other planet, the
acceleration due to gravity π P is equal to πΊ times three π naught over six π
naught all squared. We can take out a factor of three
from the numerator and a factor of six squared from the denominator, giving us three
over six squared times πΊπ naught over π naught squared. This second term, the product, is
just equal to π E, and three over six squared is equal to one twelfth. Therefore, π P is equal to one
twelfth π E. Therefore, the ratio between the
acceleration due to gravity on the other planet, π P, to the acceleration due to
gravity on Earth, π E, is equal to one to 12.

A possibly faster way to solve this
problem is to note that since the universal gravitational constant is a constant,
the acceleration due to gravity is proportional to the mass over the radius
squared. An increase by a factor of three of
the mass will result in the same increase in the acceleration due to gravity. And an increase in the radius by a
factor of six will result in a decrease in the acceleration due to gravity by a
factor of six squared.

Therefore, the acceleration due to
gravity on the other planet compared with that of Earth is multiplied by three and
then divided by six squared, which gives us one twelfth, which gives us the same
answer.