Question Video: Finding the Ratio between the Gravity of Two Planets given the Ratio between Their Masses and Radii Mathematics • Second Year of Secondary School

Video Transcript

Given that a planet’s 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 𝑎 at the surface of a uniform sphere of mass 𝑚 and radius 𝑟 is equal to 𝐺𝑚 over 𝑟 squared, where 𝐺 is the universal gravitational constant. This is the same formula as that of a particle of mass 𝑚 located at the geometric center of the sphere.

Let’s look at our two planets. Let’s say the Earth has a mass 𝑚 equals 𝑚 naught and a radius 𝑟 equals 𝑟 naught. The mass of the other planet is three times that of Earth. So its mass 𝑚 is equal to three 𝑚 naught. And since its diameter is six times that of Earth, its radius is also six times that of Earth. So 𝑟 is equal to six 𝑟 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. Let’s do this first.

So, for Earth, the acceleration due to gravity 𝑎 E is equal to 𝐺𝑚 naught over 𝑟 naught squared. And for the other planet, the acceleration due to gravity 𝑎 P is equal to 𝐺 times three 𝑚 naught over six 𝑟 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 𝐺𝑚 naught over 𝑟 naught squared. This second term, the product, is just equal to 𝑎 E, and three over six squared is equal to one twelfth. Therefore, 𝑎 P is equal to one twelfth 𝑎 E. Therefore, the ratio between the acceleration due to gravity on the other planet, 𝑎 P, to the acceleration due to gravity on Earth, 𝑎 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.