Weight and Mass

In everyday discussion many people use weight and mass to mean the same thing which is not true.

Mass is a scalar and weight is a vector. Mass is a measurement of how much matter is in an object; weight is a measurement of how hard gravity is pulling on that object. Your mass is the same wherever you are, on Earth; on the moon; floating in space, because the amount of stuff you're made of doesn't change. Your weight depends on how strong a gravitational force is acting on you at the moment; you'd weigh less on the moon than on Earth, and in space you'd weigh almost nothing at all. Mass is measured in kilograms, kg, and weight is a force and measured in newtons, N.

When you stand on a scale you are trying measure how much of you there is. People who are trying to reduce their mass hope to see the reading on the scale get smaller but they talk about losing weight. Their weight will decrease but it is because their mass is decreasing. A scale uses the persons weight to determine their mass.

You can use $\vec{F}_g = m\vec{g}$ to calculate weight.

Example: Newton's Second Law: Lifts and g

Question

A lift, with a mass of $\text{250}$ $\text{kg}$, is initially at rest on the ground floor of a tall building. Passengers with an unknown total mass, $m$, climb into the lift. The lift accelerates upwards at $\text{1.6}$ $\text{m·s$^{-2}$}$. The cable supporting the lift exerts a constant upward force of $\text{7 700}$ $\text{N}$.

Draw a labelled force diagram indicating all the forces acting on the lift while it accelerates upwards.
What is the maximum mass, m, of the passengers the lift can carry in order to achieve a constant upward acceleration of $\text{1.6}$ $\text{m·s$^{-2}$}$.

Step 1: Draw a force diagram.

We choose upwards as the positive direction.

Step 2: Gravitational force

We know that the gravitational acceleration on any object on Earth, due to the Earth, is $\vec{g}=\text{9.8}\text{ m·s$^{-2}$}$ towards the centre of the Earth (downwards). We know that the force due to gravity on a lift or the passengers in the lift will be $\vec{F}_g=m\vec{g}$.

Step 3: Find the mass, m.

Let us look at the lift with its passengers as a unit. The mass of this unit will be $(\text{250}\text{ kg} + m)$ and the force of the Earth pulling downwards (${F}_{g}$) will be $\left(\text{250}+ \text{m}\right)\times \text{9.8}\text{ m·s$^{-2}$}$. If we apply Newton's second law to the situation we get:

\begin{align*} {F}_{net}& = ma \\ {F}_{C}-{F}_{g}& = ma \\ \text{7 700}-\left(\text{250}+m\right)\left(\text{9,8}\right)& = \left(\text{250}+m\right)\left(\text{1,6}\right) \\ \text{7 700}-\text{2 500}-\text{9,8} m& = \text{400}\text{+1,6} m \\ \text{4 800}& = \text{11,4} m \\ m& = \text{421,05}\text{ kg} \end{align*}

Step 4: Quote your final answer

The mass of the passengers is $\text{421.05}$ $\text{kg}$. If the mass were larger then the total downward force would be greater and the cable would need to exert a larger force in the positive direction to maintain the same acceleration.

Tip:

In everyday use we often talk about weighing things. We also refer to how much something weighs. It is important to remember that when someone asks how much you weigh or how much an apple weighs they are actually wanting to know your mass or the apples mass, not the force due to gravity acting on you or the apple.

Weightlessness is not because there is no gravitational force or that there is no weight anymore. Weightless is an extreme case of apparent weight. Think about the lift accelerating downwards when you feel a little lighter. If the lift accelerated downwards with the same magnitude as gravitational acceleration there would be no normal force acting on you, the lift and you would be falling with exactly the same acceleration and you would feel weightless. Eventually the lift has to come to a stop.

In a space shuttle in space it is almost exactly the same case. The astronauts and space shuttle feel exactly the same gravitational acceleration so their apparent weight is zero. The one difference is that they are not falling downwards, they have a very large velocity perpendicular to the direction of the gravitational force that is pulling them towards the earth. They are falling but in a circle around the earth. The gravitational force and their velocity are perfectly balanced that they orbit the earth.

In a weightless environment, defining up and down doesn't make as much sense as in our every day life. In space this affects all sorts of things, for example, when a candle burns the hot gas can't go up because the usual up is defined by which way gravity acts. This has actually been tested.

A candle burning on earth (left) and one burning in space (right).

Weight and Mass

Weight and Mass

Example: Newton's Second Law: Lifts and g

Question

Step 1: Draw a force diagram.

Step 2: Gravitational force

Step 3: Find the mass, m.

Step 4: Quote your final answer

Tip:

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