The weight equation

Definition

Weight

The force acting on an object because of gravity, measured in newtons.

Definition

Mass

The amount of matter in an object, measured in kilograms.

The weight of an object, its mass and the gravitational field strength are linked by an equation you must recall.
In words, $strength\text{gravitational force}=\text{mass}\times\text{gravitational field strength}$ .
In symbols, the equation is $W = m g$ .
1. $W$ is the weight, or gravitational force, in newtons, $N\text{N}$ .
2. $m$ is the mass in kilograms, $kg\text{kg}$ .
3. $g$ is the gravitational field strength in newtons per kilogram, $N/kg\text{N/kg}$ .
Use $N/kgg=10\ \text{N/kg}$ at the Earth's surface unless a question gives a different value.
Convert a mass in grams into kilograms before substituting, by dividing it by $1000$ .
1. A $g250\ \text{g}$ apple has a mass of $kg0.250\ \text{kg}$ , so its weight on Earth is $N0.250\times10=2.5\ \text{N}$ .

A person with a mass of 50 kg has a weight of 490 N on Earth, 185 N on Mars and 80 N on the Moon, because the same mass is multiplied by a different gravitational field strength in each place (the Earth value here uses 9.8 N/kg rather than 10 N/kg).

Rearranging the equation

Rearranging $W = m g$ gives $m=Wgm=\dfrac{W}{g}$ , which finds a mass from a weight.
Rearranging again gives $g=Wmg=\dfrac{W}{m}$ , which finds the gravitational field strength from a weight and a mass.
The equation $W = m g$ is $F = ma$ applied to an object in free fall, with the weight as the resultant force and $g$ as the acceleration.

How the quantities are inter-related

Definition

Directly proportional

A relationship between two quantities in which multiplying one by a factor multiplies the other by the same factor.

In $W = m g$ , the weight is directly proportional to the mass when $g$ stays the same.
1. In the same place, a $kg6\ \text{kg}$ suitcase weighs three times as much as a $kg2\ \text{kg}$ suitcase.
The weight is also directly proportional to $g$ when the mass stays the same.
1. Taking an object from Earth, where $g$ is $N/kg10\ \text{N/kg}$ , to Mars, where $g$ is $N/kg3.7\ \text{N/kg}$ , multiplies its weight by $0.37$ .
For a fixed weight, mass and $g$ are inversely proportional, so a place with half the field strength needs twice the mass to give the same weight.
1. A weight of $N1600\ \text{N}$ comes from a mass of $kg160\ \text{kg}$ on Earth but from a mass of $kg1000\ \text{kg}$ on the Moon, where $g$ is $N/kg1.6\ \text{N/kg}$ .
The ratio of one object's weights in two places equals the ratio of the field strengths, so $W1W2=g1g2\dfrac{W_1}{W_2}=\dfrac{g_1}{g_2}$ $\frac{W _{1}}{W _{2}} = \frac{g _{1}}{g _{2}}$ .
1. The Moon's field strength is about one sixth of the Earth's, so any object on the Moon weighs about one sixth of its weight on Earth.

Gravitational potential energy

Definition

Gravitational potential energy

The energy held in the gravitational store of an object because of its height in a gravitational field.

The gravitational field strength also appears in the equation for the energy in an object's gravitational store, which you must recall.
In words, $strength×height\text{gravitational potential energy}=\text{mass}\times\text{gravitational field strength}\times\text{height}$ .
In symbols, the equation is $E_{p} = m g h$ .
1. $E_p$ is the gravitational potential energy in joules, $J\text{J}$ .
2. $m$ is the mass in kilograms, $g$ is in newtons per kilogram, and $h$ is the height in metres.
Lifting an object through a height $h$ at a steady speed needs a force equal to its weight, $m g$ , so the work done, and the energy gained by the gravitational store, is $mg×hmg\times h$ .
For the same mass and height, the energy is directly proportional to $g$ , so lifting a box $m1\ \text{m}$ on the Moon needs about one sixth of the energy needed on Earth.

Example

Weight of a parcel

A parcel has a mass of $kg3.2\ \text{kg}$ and sits on a shelf at the Earth's surface.

W=mg=3.2\ \text{kg}\times10\ \text{N/kg}=32\ \text{N}

The weight of the parcel is $N32\ \text{N}$ .

Example

Mass from a weight

A crate is found to weigh $N450\ \text{N}$ on Earth.

m=\frac{W}{g}=\frac{450\ \text{N}}{10\ \text{N/kg}}=45\ \text{kg}

The mass of the crate is $kg45\ \text{kg}$ .

Example

Weight on the Moon from a weight on Earth

An astronaut's spacesuit weighs $N1200\ \text{N}$ on Earth, and the gravitational field strength on the Moon is $N/kg1.6\ \text{N/kg}$ .

m=\frac{W}{g}=\frac{1200\ \text{N}}{10\ \text{N/kg}}=120\ \text{kg}

W_{\text{Moon}}=mg_{\text{Moon}}=120\ \text{kg}\times1.6\ \text{N/kg}=192\ \text{N}

The spacesuit has the same mass of $kg120\ \text{kg}$ on the Moon, but its weight there is only $N192\ \text{N}$ .

Example

Field strength of another planet

A $kg2.0\ \text{kg}$ test mass on a space probe's arm weighs $N7.4\ \text{N}$ on the surface of a planet.

g=\frac{W}{m}=\frac{7.4\ \text{N}}{2.0\ \text{kg}}=3.7\ \text{N/kg}

The gravitational field strength at the planet's surface is $N/kg3.7\ \text{N/kg}$ , which matches the value for Mars.

Example

Energy gained by a lifted bag

A $kg0.60\ \text{kg}$ school bag is lifted from the floor onto a desk $m1.5\ \text{m}$ higher.

E_p=mgh=0.60\ \text{kg}\times10\ \text{N/kg}\times1.5\ \text{m}=9.0\ \text{J}

The gravitational store of the bag gains $J9.0\ \text{J}$ .

Common Mistake

Do not substitute a mass in grams into $W = m g$ , because the mass must be in kilograms.
Do not change the mass when an object moves to another planet, because only its weight changes.
Do not confuse the gravitational field strength, $g$ , with the abbreviation $g\text{g}$ for grams.

Exam technique

Calculations with gravitational field strength

Write $W = m g$ or $E_p=mgh$ from memory before substituting, because recalling these equations is part of what is assessed.
In a two-step question, find the mass from the weight in one place first, then use it with the new value of $g$ .
When asked how one quantity changes as another changes, use proportional reasoning, such as " $g$ is halved so the weight is halved", and support it with numbers.
Give every answer with its unit, using newtons for weight, kilograms for mass and newtons per kilogram for field strength.

Self review

Write the equation that links weight, mass and gravitational field strength.
Calculate the weight on Earth of a $g750\ \text{g}$ book.
Calculate the mass of an object that weighs $N85\ \text{N}$ on Earth.
Describe what happens to an object's weight on a planet where the gravitational field strength is three times larger than on Earth.
Calculate the gravitational potential energy gained by a $kg2.0\ \text{kg}$ bag lifted through $m1.2\ \text{m}$ .

The weight equation

Definition

Weight

The force acting on an object because of gravity, measured in newtons.

Definition

Mass

The amount of matter in an object, measured in kilograms.

The weight of an object, its mass and the gravitational field strength are linked by an equation you must recall.
In words, $strength\text{gravitational force}=\text{mass}\times\text{gravitational field strength}$ .
In symbols, the equation is $W = m g$ .
1. $W$ is the weight, or gravitational force, in newtons, $N\text{N}$ .
2. $m$ is the mass in kilograms, $kg\text{kg}$ .
3. $g$ is the gravitational field strength in newtons per kilogram, $N/kg\text{N/kg}$ .
Use $N/kgg=10\ \text{N/kg}$ at the Earth's surface unless a question gives a different value.
Convert a mass in grams into kilograms before substituting, by dividing it by $1000$ .
1. A $g250\ \text{g}$ apple has a mass of $kg0.250\ \text{kg}$ , so its weight on Earth is $N0.250\times10=2.5\ \text{N}$ .

Rearranging the equation

Rearranging $W = m g$ gives $m=Wgm=\dfrac{W}{g}$ , which finds a mass from a weight.
Rearranging again gives $g=Wmg=\dfrac{W}{m}$ , which finds the gravitational field strength from a weight and a mass.
The equation $W = m g$ is $F = ma$ applied to an object in free fall, with the weight as the resultant force and $g$ as the acceleration.

How the quantities are inter-related

Definition

Directly proportional

A relationship between two quantities in which multiplying one by a factor multiplies the other by the same factor.

In $W = m g$ , the weight is directly proportional to the mass when $g$ stays the same.
1. In the same place, a $kg6\ \text{kg}$ suitcase weighs three times as much as a $kg2\ \text{kg}$ suitcase.
The weight is also directly proportional to $g$ when the mass stays the same.
1. Taking an object from Earth, where $g$ is $N/kg10\ \text{N/kg}$ , to Mars, where $g$ is $N/kg3.7\ \text{N/kg}$ , multiplies its weight by $0.37$ .
For a fixed weight, mass and $g$ are inversely proportional, so a place with half the field strength needs twice the mass to give the same weight.
1. A weight of $N1600\ \text{N}$ comes from a mass of $kg160\ \text{kg}$ on Earth but from a mass of $kg1000\ \text{kg}$ on the Moon, where $g$ is $N/kg1.6\ \text{N/kg}$ .
The ratio of one object's weights in two places equals the ratio of the field strengths, so $W1W2=g1g2\dfrac{W_1}{W_2}=\dfrac{g_1}{g_2}$ $\frac{W _{1}}{W _{2}} = \frac{g _{1}}{g _{2}}$ .
1. The Moon's field strength is about one sixth of the Earth's, so any object on the Moon weighs about one sixth of its weight on Earth.

Gravitational potential energy

Definition

Gravitational potential energy

The energy held in the gravitational store of an object because of its height in a gravitational field.

The gravitational field strength also appears in the equation for the energy in an object's gravitational store, which you must recall.
In words, $strength×height\text{gravitational potential energy}=\text{mass}\times\text{gravitational field strength}\times\text{height}$ .
In symbols, the equation is $E_{p} = m g h$ .
1. $E_p$ is the gravitational potential energy in joules, $J\text{J}$ .
2. $m$ is the mass in kilograms, $g$ is in newtons per kilogram, and $h$ is the height in metres.
Lifting an object through a height $h$ at a steady speed needs a force equal to its weight, $m g$ , so the work done, and the energy gained by the gravitational store, is $mg×hmg\times h$ .
For the same mass and height, the energy is directly proportional to $g$ , so lifting a box $m1\ \text{m}$ on the Moon needs about one sixth of the energy needed on Earth.

Example

Weight of a parcel

A parcel has a mass of $kg3.2\ \text{kg}$ and sits on a shelf at the Earth's surface.

W=mg=3.2\ \text{kg}\times10\ \text{N/kg}=32\ \text{N}

The weight of the parcel is $N32\ \text{N}$ .

Example

Mass from a weight

A crate is found to weigh $N450\ \text{N}$ on Earth.

m=\frac{W}{g}=\frac{450\ \text{N}}{10\ \text{N/kg}}=45\ \text{kg}

The mass of the crate is $kg45\ \text{kg}$ .

Example

Weight on the Moon from a weight on Earth

An astronaut's spacesuit weighs $N1200\ \text{N}$ on Earth, and the gravitational field strength on the Moon is $N/kg1.6\ \text{N/kg}$ .

m=\frac{W}{g}=\frac{1200\ \text{N}}{10\ \text{N/kg}}=120\ \text{kg}

W_{\text{Moon}}=mg_{\text{Moon}}=120\ \text{kg}\times1.6\ \text{N/kg}=192\ \text{N}

The spacesuit has the same mass of $kg120\ \text{kg}$ on the Moon, but its weight there is only $N192\ \text{N}$ .

Example

Field strength of another planet

A $kg2.0\ \text{kg}$ test mass on a space probe's arm weighs $N7.4\ \text{N}$ on the surface of a planet.

g=\frac{W}{m}=\frac{7.4\ \text{N}}{2.0\ \text{kg}}=3.7\ \text{N/kg}

The gravitational field strength at the planet's surface is $N/kg3.7\ \text{N/kg}$ , which matches the value for Mars.

Example

Energy gained by a lifted bag

A $kg0.60\ \text{kg}$ school bag is lifted from the floor onto a desk $m1.5\ \text{m}$ higher.

E_p=mgh=0.60\ \text{kg}\times10\ \text{N/kg}\times1.5\ \text{m}=9.0\ \text{J}

The gravitational store of the bag gains $J9.0\ \text{J}$ .

Common Mistake

Do not substitute a mass in grams into $W = m g$ , because the mass must be in kilograms.
Do not change the mass when an object moves to another planet, because only its weight changes.
Do not confuse the gravitational field strength, $g$ , with the abbreviation $g\text{g}$ for grams.

Exam technique

Calculations with gravitational field strength

Write $W = m g$ or $E_p=mgh$ from memory before substituting, because recalling these equations is part of what is assessed.
In a two-step question, find the mass from the weight in one place first, then use it with the new value of $g$ .
When asked how one quantity changes as another changes, use proportional reasoning, such as " $g$ is halved so the weight is halved", and support it with numbers.
Give every answer with its unit, using newtons for weight, kilograms for mass and newtons per kilogram for field strength.

Self review

Write the equation that links weight, mass and gravitational field strength.
Calculate the weight on Earth of a $g750\ \text{g}$ book.
Calculate the mass of an object that weighs $N85\ \text{N}$ on Earth.
Describe what happens to an object's weight on a planet where the gravitational field strength is three times larger than on Earth.
Calculate the gravitational potential energy gained by a $kg2.0\ \text{kg}$ bag lifted through $m1.2\ \text{m}$ .

2.3.10 Relating force, mass and gravitational field strength

The weight equation

Rearranging the equation

How the quantities are inter-related

Gravitational potential energy

Weight of a parcel

Mass from a weight

Weight on the Moon from a weight on Earth

Field strength of another planet

Energy gained by a lifted bag

Calculations with gravitational field strength

Review 2.3.10 Relating force, mass and gravitational field strength by teaching Genie

2.3.10 Relating force, mass and gravitational field strength

The weight equation

Rearranging the equation

How the quantities are inter-related

Gravitational potential energy

Weight of a parcel

Mass from a weight

Weight on the Moon from a weight on Earth

Field strength of another planet

Energy gained by a lifted bag

Calculations with gravitational field strength

Review 2.3.10 Relating force, mass and gravitational field strength by teaching Genie

2.3.10 Relating force, mass and gravitational field strength

The weight equation

Rearranging the equation

How the quantities are inter-related

Gravitational potential energy

Weight of a parcel

Mass from a weight

Weight on the Moon from a weight on Earth

Field strength of another planet

Energy gained by a lifted bag

Calculations with gravitational field strength

Review 2.3.10 Relating force, mass and gravitational field strength by teaching Genie

2.3.10 Relating force, mass and gravitational field strength Revision Guide

2.3.10 Relating force, mass and gravitational field strength

The weight equation

Rearranging the equation

How the quantities are inter-related

Gravitational potential energy

Weight of a parcel

Mass from a weight

Weight on the Moon from a weight on Earth

Field strength of another planet

Energy gained by a lifted bag

Calculations with gravitational field strength

Review 2.3.10 Relating force, mass and gravitational field strength by teaching Genie

2.3.10 Relating force, mass and gravitational field strength Revision Guide