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If two objects with different masses are subject to the same acceleration due to gravity the object with more mass will also weigh more. Mass is the amount of matter an object contains and weight is the force it exerts due to gravity. Weight is (mass)x(acceleration due to gravity), or mg.
Objects on the surface of the earth all experience about the same acceleration due to gravity of approximately 9.8 meters/second^2. This value varies only slightly with location on the earth due to irregularities in the earth's shape and topography. We tend to use the words mass and weight interchangeably in regards to the quantity of something on earth. We measure body weight rather than mass because the devices with which we "weigh" ourselves measure the force we exert. The mass of an object is the same everywhere but weight varies based on gravity. For example, an object's weight on the moon is about 1/6 what is is on Earth.
Mass is defined as the amount of matter within an object. Mass is measured in grams. Weight, on the other hand, is defined as a measurement of the force of gravity on an object. Weight is measured in Newtons.
The mass of an object does not change. For example, a person that has a mass of 100 kg on Earth will have a mass of 100 kg on the moon. However, weight will change as the force of gravity changes. Therefore, the same person will weigh less on the moon as they do on Earth.
When two objects are both on the same planet (and experiencing the same acceleration due to gravity), the gravitational force on the objects increases as the mass of the objects increases. Thus, the object that has a greater mass will also have a greater weight.
The link that is provided below explain the above concepts in more detail.
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