Weight and mass

Problem

What is the difference between weight and mass? What would a spring scale show when weighing the same object first on Earth and then on the Moon? And a two-pan balance?

Solution

Mass is the amount of matter in a body, measured in kg, and is an invariant property: it does not change when the object is moved from one celestial body to another.

Weight is the force with which a celestial body attracts the object, measured in newtons and given by: $P = m\,g$ It therefore depends on the local acceleration due to gravity $g$, which differs from body to body.

A spring scale measures the weight force directly (the spring’s extension). With the same object of mass $m$: $P_\text{Earth} = m\,g_\text{Earth}, \quad g_\text{Earth}\approx 9{,}81~\text{m/s}^2 \qquad P_\text{Moon} = m\,g_\text{Moon}, \quad g_\text{Moon}\approx 1{,}62~\text{m/s}^2$ On the Moon it would read a weight about $6$ times smaller: $\frac{P_\text{Moon}}{P_\text{Earth}} = \frac{g_\text{Moon}}{g_\text{Earth}} = \frac{1{,}62}{9{,}81} \approx 0{,}165$ A two-pan balance, instead, compares the unknown mass with reference masses: both pans feel the same $g$, which cancels out in the equilibrium. It would therefore give the same value on Earth and on the Moon. $\ev{\text{invariant mass};\ \text{spring scale: } P_\text{Moon}\approx P_\text{Earth}/6;\ \text{two-pan balance: same value}}$

Links

Topics: Gravitation Concepts: Weight force Skills: Symbolic set-up