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What terminology is used to refer to weights/ mass/ weight of mass/ mass of weights when referring to the mass of weights in a physics report? My question is more of the weights that we use in the physics laboratory to hang on springs/ rulers, do we refer to them as masses or weights?

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We use the term mass, when we mean the mass of a weight, and we use the term weight, when we mean the weight of a mass. :-)

The important thing to remember is, that the mass is the same everywhere, while the weight varies with the local gravity. So if you are referring to the constant mass of an object, you use mass expressed in kg. If, however, you mean the force of gravity on that object, you talk about weight and measure it in Newton.

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If you care about the inertia you use "mass". When you are considering the force of gravity you use "weight".

So when you do calculations about the force in the cables due to acceleration of an elevator car you need to know both it's mass and it's weight...

The (calibrated) object you place on a scale is called a "weight" - because that is the property you most care about. But a calibrated weight is never calibrated in Newtons - it is calibrated in grams. This is because weight is a function of position on earth - mass is not. So when you want to determine the mass of an unknown object you can use scales to compare its weight with that of a known object. When you know the ratio of weights you know the ratio of masses. Thus it is OK to calibrate a scale in grams and indicate mass in grams - as long as you understand that you had to calibrate locally using a known mass to get the conversion factor.

This is one reason why even cheap 5-digit digital balances (you can get them at Amazon for about $30) ship with calibration weights - to get that 0.01% accuracy you need a local reference calibration. There is about a 0.7% change in force of gravity from North Pole to the Nevado Huescaràn mountain in Peru (you are heavier at the North Pole) - or about 0.01% for every degree of latitude (on average).

In most other contexts when you talk about a physical object you call it a "mass".

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