Gravity keeps us with our feet on the grounds: you can calculate the acceleration due to gravity, a quantity defining the feeling of weight, the speed of falling objects, and many more things surprisingly quickly. Keep reading: you'll learn the formula for gravitational acceleration, how to calculate the acceleration due to gravity on the Moon, and much more!

What is gravity? Introduction to the calculations for the gravity of a planet

Gravity is one of the four fundamental forces that keep our Universe together. While other forces act mainly on subatomic particles, gravity dominates Nature on an astronomical scale at a subatomic level: planets, stars, and galaxies communicate and interact through this force.

Now, thanks to Einstein, we know that gravity is a manifestation of mass as a curvature in space-time. The more mass an object has, the deeper the indentation it causes in space-time. This change in curvature attracts other massive objects: they experience the effect of the gravitational field.

Our understanding of gravity, however, is slightly different due to our everyday perception. Things fall to the ground thanks to gravity, and our feet stick to Earth thanks to it. We feel gravity as a constant force in the "vertical" direction. Only by lifting our eyes, we can appreciate it in other forms, but then again: it took centuries to understand it (thanks, Newton); it's not that immediate!

What happens is that we are probes of Earth's gravitational field. At least, probes of a tiny part of it: unless you are an astronaut, you ventured at most 10 km10\ \text{km} above our planet's surface. Consequently, you experience an almost constant gravitational acceleration: the formula we'll see in the next section will explain why.

How to find the acceleration due to gravity: calculate gravity acceleration for any massive body

If you want to learn how to find the acceleration due to gravity, you need to take a step back and learn how to calculate the gravitational force. To do so, we ask for Newton's help.

We know how to calculate the gravitational force:

FMm=GMmr2F_{\text{Mm}} = G\cdot \frac{M\cdot m}{r^2}

Where:

  • FMmF_{\text{Mm}} — The gravitational force between two massive bodies M\text{M} and m\text{m};
  • MM and mm — The masses of the bodies;
  • GG — The gravitational constant; and
  • rr — The **distance between the centers of mass of the objects.

Let's invoke Newton once again. As we explained thoroughly in our Newton's second law calculator, there is a simple relationship between force and acceleration: this holds for the acceleration due to gravity, on Earth, and everywhere else in the universe (as far as we know).

Newton's second law of motion's formula for the acceleration due to gravity is:

FMm=mgF_{\text{Mm}} = m\cdot g

Where:

  • FMmF_{\text{Mm}} — The gravitational force as seen above;
  • mm — The mass of the probing body; and
  • gg — The acceleration due to gravity.

How do you calculate the acceleration due to gravity? Begin by comparing the results of the previous two formulas, for the gravitational acceleration and the gravitational force:

GMmr2=mgG\cdot \frac{M\cdot m}{r^2} = m\cdot g

As you can see, the mass of the probing body, mm, appears on both sides. Erase it: you will isolate the value of gg: you found the formula for the acceleration due to gravity!

g=GMr2g =G\cdot \frac{M}{r^2}

🙋 The value of GG, thr gravitational constant, is G=6.6743 m3/kgs2G= 6.6743\ \text{m}^3/\text{kg}\cdot\text{s}^2.

Our magnitude of acceleration calculator can help you understand these calculations even better!

A few words on the formula for the acceleration due to gravity

You know how to find the acceleration due to gravity: it was not that complicated, right? The formula in itself is rather unassuming. However, we can notice a couple of exciting things:

  • The effect of distance on the acceleration due to gravity: in the formula for the calculations of the g acceleration, we find the distance squared at the denominator. Gravity has a strong dependency on distance, and by the way, this is why you should never trust astrology: Saturn is way too far to affect you with its mass!
  • The absence of the probing mass in the equation for the acceleration due to gravity. Your mass or the mass of any other object doesn't affect the value of the g-acceleration experienced. Notice that it would matter in case of an analysis of the gravitational force: however, it often happens that we consider bodies with a vast difference in mass, and we can neglect the probe. Remember that we are not making the same assumption here!

Calculate the acceleration due to gravity on the Moon and on Earth.

Let's calculate the acceleration due to gravity on Earth. We will find the well-known value we experience every day of our life.

Let's substitute the values for Earth's mass in the formula and the distance. But which distance? In the gravity acceleration formula, we use the distance between the center of mass of the objects in analysis: in our example, it corresponds to the value of Earth's radius. Let's try our hand at calculating the g acceleration:

g=GMr2=6.6743 ⁣ m3kgs25.972 ⁣× ⁣1024 kg6, ⁣371, ⁣000 m=9.82 ms2\begin{align*} g_{\oplus} &= G\cdot\frac{M_{\oplus}}{r_{\oplus}^2} \\ \\ &=6.6743\!\ \small{\frac{\text{m}^3}{\text{kg}\cdot\text{s}^2}}\frac{5.972\! \times\! 10^{24}\ \small{\text{kg}}}{6,\!371,\!000\ \small{\text{m}}} \\ \\ & =9.82\ \frac{\text{m}}{\text{s}^2} \end{align*}

This result is familiar! The value of about 9.81 m/s29.81\ \text{m}/\text{s}^2 is the acceleration every object on Earth's surface experience daily. You can find out more about this value at our [g force calculator].(calc:5450)

We calculated a planet's gravity; what about a moon's gravity? What about the Moon's gravity? To calculate the acceleration due to gravity on the Moon, input in our acceleration due to gravity calculator the following values:

  • M=7.348×1022 kgM_{☾} = 7.348 \times 10^{22}\ \text{kg} — The Moon's mass; and
  • 1737.4 km1737.4\ \text{km} — The Moon's radius.

The result of the formula for the gravity acceleration is g=1.632 m/s2g_{☾} = 1.632\ \text{m}/\text{s}^2. The acceleration due to gravity on the Moon is six times lower than the one on Earth.

Oh, and by the way, if we substitute the radius of the Moon's orbit around the Earth, we can calculate the acceleration due to gravity experienced by the Moon: g=2.69×103 m/s2g_{\oplus}=2.69\times 10^{-3}\ \text{m}/\text{s}^2. What about artificial satellites? First, calculate their orbit around Earth, then use our calculator for the g acceleration!

Davide Borchia
g = GM/R²
Mass (M)
lb
Radius (R)
mi
Acceleration due to gravity (g)
ft/s²
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