If we're talking about gravity on the surface of the Earth, then we use the radius of the Earth as the value r, which is generally referenced as RE (and has a value of 6.38 x 106 m). The weight of an object on the Earth's surface is, therefore, given by the equation on the right.
Weight vs. MassIt is easy to confuse mass and weight, because most of the world uses the metric convention of measuring mass instead of weight. We Americans, therefore, get confused over the difference between kilograms and pounds, even though these measure different quantities. Pounds are a measure of the gravitational force on us and it changes when we, for example, travel to the moon, where we use the moon's mass instead of the Earth's to calculate weight. Mass only changes if we go on a diet and/or exercise. A change in mass will result in a change in weight, however, and you can determine the mass of an object by measuring it's weight and applying the aforementioned gravity formula to obtain its mass.
Acceleration Due to GravityNewton's Laws of Motion tell us that force is proportional to the product of mass and acceleration. Therefore, we can divide both sides of the equation by the individual object's mass to obtain an equation for the acceleration caused by gravity, g. (To view this equation, click on the image to the right to move on to the second image.)
Since we know all of the values on the right side of this equation, we obtain a constant gravitational acceleration on the surface of the Earth, which comes out to 9.80 m/s2 (note that here m stands for the unit meters, not a variable for mass).
These equations assume that you're at or relatively near the Earth's surface. Obviously, the further away you get, the less accurate RE is as an approximation. The required level of precision for your work will determine if you can use this value for g or whether you will need to calculate it directly with a more specific value of r.
Shortcut to Weight & ForceIn most physics problems in the classroom, we refer to the gravitational force as simply the product of mass and gravitational acceleration, Fg = mg. Similarly, since weight is a measurement of the force of gravity, w = mg.
Again, this does not work in cases where approximating the location as the Earth's surface doesn't obtain the proper level of precision. But 99% of the time, if your problem takes place on the Earth, you will be using this formula to obtain the force of gravity (or weight). In these cases, you'll have to take account exact position to get your precise value, using the more comprehensive theory of universal gravitation directly.
Gravitational Potential EnergyMany problems involving gravity can be solved by applying the concept of gravitational potential energy. Since potential energy is a relative quantity, and problems generally require knowing the change in the potential energy rather than the absolute potential energy, the equation for gravitational potential energy U near the Earth's surface is straightforward. It is defined in term of the object's mass m, the acceleration of gravity g, and the distance y above the coordinate origin (generally the ground) in the form:
U = mgyThis equation can be applied in conjunction with kinetic energy to determine how one quantity changes if the total energy remains the same.