Because the mass of the flask with water exceeds the amount that can be weighed on the digital balances, the triple-beam-balance is used, which only weighs to the nearest 0.01 g. Why doesn't using this balance affect the number of significant digits to which the experimental molecular weight of carbon dioxide can be reported

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Mass is the amount of matter in an object.  weight is the measurement of the pull of gravity on an object.  say for example, you travel to the moon which has a different gravitational field compared to the gravitational field of earth. (gravitational fields: earth = 1, moon = 0.165)  now when you are on the moon, the pull of gravity on you is different than on the earth. and as weight is the measurement of the pull of gravity on an object, that means yes, your weight can and will change on the moon.  however, on the moon, on earth or even on jupiter, the amount of mass inside of you remains the same. meaning your physical mass remains constant.  in conclusion it is possible for your weight to change as your mass remains constant. 

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