Calculate Surface Gravity

This activity will illustrate how to calculate the surface gravity of planets, satellites and the Sun.

Time requirement: 50 minutes as an activity for older or mathematically advanced students.

Materials

Procedures

You have learned that the surface gravity (g) of a body depends on the mass (M) and the radius (r) of the given body. The formula which relates these quantities is:

g = G * M / r2

where G is called the Gravitational constant. Remember that the notation r2 means r to the 2nd power, or r squared.

You will calculate the surface gravity for a number of bodies using the MKS system where the units for distance are meters, the units for mass are kilograms, and the units for time are seconds. In this system, the gravitational constant has the value:

G = 6.67 * 10-11 Newton-meter2/kilogram2.

As an example, the mass M of the Earth is 5.98 * 1024 kilograms. The radius r of the Earth is 6378 kilometers, which is equal to 6.378 * 106 meters. The surface gravity on Earth can therefore be calculated by:

g = G * M / r2
= (6.67 * 10-11) * (5.98 * 1024) / (6.378 * 106)2
= 9.81 meters/second2

So, a simple formula from the science of Physics can be used to calculate the surface gravity for a body (in this case the Earth) if you know the mass of the body and its radius! The assumption in using this formula is that the body is spherical, but this is a pretty good assumption. If the radii of a body at its equator and pole are very different, then the surface gravity is different at those places and should be calculated separately.

The surface gravity for the Earth is therefore 9.81 meters per second2, or 9.81 meters per second per second. This is the acceleration due to gravity that an object feels near the surface of the Earth. For example, if an object were dropped from rest near the Earth's surface, it would accelerate to a velocity of 9.81 meters per second after one second, and the velocity would increase by another 9.81 meters per second for every additional second that the object was falling (in the vicinity of the Earth's surface).

A table of masses and radii is given below for many bodies in the Solar System. Make sure to convert the radii from kilometers to meters when making the calculation, and make sure that you can calculate the surface gravity of the Earth correctly. Then, calculate the surface gravity at each of the other bodies. Think about how much you would weigh on the surface of these bodies relative to how much you weigh on the surface of the Earth.

Body Mass (kg) Radius (km)
Earth 5.98 * 1024 6378
Mercury 3.30 * 1023 2439
Venus 4.87 * 1024 6051
Mars 6.42 * 1023 3393
Jupiter 1.90 * 1027 71492
Saturn 5.69 * 1026 60268
Uranus 8.68 * 1025 25559
Neptune 1.02 * 1026 24764
Pluto 1.29 * 1022 1150
Moon 7.35 * 1022 1738
Ganymede 1.48 * 1023 2631
Titan 1.35 * 1023 2575
Sun 1.99 * 1030 696000

Note that the Gas Giant planets (Jupiter, Saturn, Uranus and Neptune) do not have solid surfaces. The radii of these planets are specified at the point where the pressure in their atmospheres is approximately equal to that at the surface of the Earth. Ganymede is the largest moon of Jupiter and Titan is the largest satellite of Saturn. These are the two largest moons in the Solar System. Note also how the Sun is so much larger and more massive than all the other bodies in the Solar System.


Last updated: February 05, 1998
Joe Twicken / joe@nova.stanford.edu
Rob Wigand