Time requirement: 50 minutes as an activity for older or mathematically advanced students.
g = G * M / r^{2}
where G is called the Gravitational constant. Remember that the notation r^{2} 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-meter^{2}/kilogram^{2}.
As an example, the mass M of the Earth is 5.98 * 10^{24} kilograms. The radius r of the Earth is 6378 kilometers, which is equal to 6.378 * 10^{6} meters. The surface gravity on Earth can therefore be calculated by:
g | = | G * M / r^{2} |
= | (6.67 * 10^{-11}) * (5.98 * 10^{24}) / (6.378 * 10^{6})^{2} | |
= | 9.81 meters/second^{2} |
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 second^{2}, 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 * 10^{24} | 6378 |
Mercury | 3.30 * 10^{23} | 2439 |
Venus | 4.87 * 10^{24} | 6051 |
Mars | 6.42 * 10^{23} | 3393 |
Jupiter | 1.90 * 10^{27} | 71492 |
Saturn | 5.69 * 10^{26} | 60268 |
Uranus | 8.68 * 10^{25} | 25559 |
Neptune | 1.02 * 10^{26} | 24764 |
Pluto | 1.29 * 10^{22} | 1150 |
Moon | 7.35 * 10^{22} | 1738 |
Ganymede | 1.48 * 10^{23} | 2631 |
Titan | 1.35 * 10^{23} | 2575 |
Sun | 1.99 * 10^{30} | 696000 |