Will two objects with the same mass have the same weight, and why or why not?
Two objects with the same mass will have the same weight only if they are in the same gravitational field. Weight is the gravitational force on an object, given by $W = mg$, so it depends on the local acceleration due to gravity $g$. If $g$ changes (for example, on Earth vs. the Moon, or at different altitudes), their weights change even though their masses stay the same.
What you are comparing
Mass is the amount of matter in an object (measured in kg) and does not change just because you move the object. Weight is a force (measured in newtons, N) caused by gravity pulling on that mass.
The relationship between mass and weight
The key equation is: $$W = mg$$ where $W$ is weight, $m$ is mass, and $g$ is the local gravitational field strength.
When equal masses do have equal weights
If two objects have the same mass ($m$ is the same) and they are in the same place so $g$ is the same, then $W$ is the same for both.
When equal masses do not have equal weights
If the objects are in different gravitational fields, $g$ differs, so their weights differ even though their masses match.
Example: if each has mass $1\text{ kg}$,
- On Earth, $W \approx 1 \cdot 9.8 = 9.8\text{ N}$
- On the Moon, $W \approx 1 \cdot 1.6 = 1.6\text{ N}$
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