Well first and foremost, we can see that the problem gives us the mass in grams, which is a metric unit. And as such, we will want to solve for the metric unit of weight, the Newton.
In order to solve for weight, one must first realize what weight is. Weight is simply the force that the earth (or any given celestial being the object resides on) is applying to the object, to keep it pulled down. So weight is force.
Newton's second law of motion states that Force is equal to mass times acceleration. Mass is 1,000 grams, but one must keep in mind that a Newton is a kilogram times meter per second^2, so we want to convert our mass to kilograms. 1,000 grams is equal to 1 kilogram. Now we have our mass.
Next is acceleration. As stated earlier, weight is the force that gravity applies on an object, so acceleration must then be the acceleration due to gravity, which we know is 9.8 meters per second^2. Now we just plug in numbers.
F=ma
F=(1 kg)(9.8 m/sec^2)
F=9.8 (kg x m)/sec^2
F=9.8 N
Hence the earth is applying 9.8 Newtons of force on the object due to gravity, so the object weighs 9.8 Newtons
Hope this helps :)
