Technically, weight is the force which an object applies in downward direction, with Newton as its SI unit. However in non-technical usage, weight is a measure of heaviness or lightness of an object, measured mostly in kgs/grams. Is the non-technical definition of weight of an object same as the mass of that object? Or is the non-technical weight also affected by gravity?
Gravity – Is the Non-Technical Definition of Weight the Same as Mass or Affected by Gravity?
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Related Solutions
The mass of the object always stays the same. The balance can only measure the downward force exerted on it by the bob. The force measured by the balance is simply the weight of the masses on one side needed to balance the downward force of the bob on the other side.
In air, the only appreciable force will be the downward force from gravity, aka the weight of the bob. In water, there is also a significant upward force due to the buoyant force exerted on the bob by the water. So in the water, the balance is measuring the difference between the weight of the bob and the buoyant force. The relevant physics and formulas can all be found on Wikipedia easily. If you define the (true) specific gravity $S$ as the ratio of the density of your bob $\rho_B$ to the density of water, i.e. $S = \frac{\rho_B}{\rho_{H_20}}$, you should be able to show that $$ \frac{f_{water}}{f_{air}} = 1 - \frac{1}{S},$$ where $f_{air}$ and $f_{water}$ are the forces measured by your balance in air and in water respectively.
Regarding the use of grams or Newtons, they are often used interchangeably to talk about the weight of an object, although this is technically rather sloppy because they are not the same thing in general. The two units measure fundamentally different things, one is a mass and one is a force. However, since all objects on the Earth are subject to the same acceleration $g$ due to gravity, there is a natural way to change between one and the other, by the formula $f = mg$. Whenever people use grams to measure forces, or Newtons to measure mass, it is this correspondence that they are implicitly referring to.
There's a thing called a "slug". " It is a mass that accelerates by 1 ft/s2 when a force of one pound-force (lbF) is exerted on it." (wikipedia).
Sometimes you'll see reference to a "pound-mass" to indicate a mass which weighs one pound at sea level (on Earth, thank you! :-) ).
The problem is that pounds and kilograms have been used colloquially since forever to describe the weight of objects. Scientific usage differs from informal usage such as "shipping weight".
Best Answer
While it is true that sometimes weight and mass are referred to both with the same units (kg, grams etc.) this is technically incorrect (there really is no "technical versus non-technical definitions", but you are right in that the terms are frequently misused).
Mass and weight are two distinct quantities. Mass is a measure of the total amount of matter in a object, and a measure of the degree to which the object will resist a force. It is a scalar quantity. If we let $m$ represent the mass of an object, then the weight it has is given by $${\bf W}=m{\bf g}$$ where $\bf g$ is the strength of the earth's gravitational field. Weight is a force$^1$ and its units are Newtons, and it is a vector quantity.
If you weighed yourself on a scale and got a reading of say $70 kg$, then this is your mass (yes, they are designed to output your mass and not weight by its true definition). Your weight is therefore $$W=70\times 9.8\approx 700 \ \text{Newton}$$
$^1$ To illustrate that weight is indeed a force, note that $$g=\frac{GM}{r_E^2}$$ meaning $$W=mg=\frac{GMm}{r_E^2}$$ which is also Newton's law for gravitational force ($M$ is the mass of the earth, $G$ is the gravitational constant and $r_E$ is earth's radius).