Forces
Have you ever wondered:
What’s it like to walk on the moon?
or
Why does my stomach lurch in a lift?
The answers are both linked to gravity
and the following activities will help to
answer these questions and others.
Before we start...
Two words that we need to distinguish between
are ‘mass’ and ‘weight’.
In everyday language people talk about:
• weighing ingredients
• their own weight
• the weight of a baby
In each of these cases they are actually talking
about ‘mass’, which is measured in kilograms.
Weight?... we’ll get to that...
What happens?
Take two objects of differing masses,
a pen and a book would work well.
One person hold them at exactly the same height.
Everyone watch and listen carefully...
Release the two objects at exactly the same time.
What happened?
What happened?
How many noises did you hear?
What does this tell you?
There is an equation linking initial velocity (u), time
(t), displacement (s), and acceleration (a).
s = ut + ½at2
What can you say about the acceleration for each
object?
This acceleration is called the acceleration
due to gravity.
A bit of history
Galileo Galilei (1564-1642) is said to have
done essentially this same experiment,
dropping balls of the same material but
different masses from the Tower of Pisa.
He concluded that they reached the ground
at the same time. Until then, it was
believed that heavier objects accelerated
faster.
Historical note: there is some doubt about whether or not it
was Pisa, and even whether or not it was Galileo, but it was
around this time that Aristotle’s belief was being questioned.
What is Gravity?
Isaac Newton (1642-1726) devised a
theory that between any two bodies is
an attractional force which is related to
each of their masses and to the
distance between them.
The bigger their masses, the stronger the attraction.
The closer they are, the stronger the attraction.
What is Gravity?
Newton observed the planets, whose
motion seemed to suggest the
following formula, known as Newton’s
Law of Universal Gravitation:
𝑚1 𝑚2
F=G
𝑟2
F is the force between the two objects (N)
m1 and m2 are the masses (kg) of the two objects
r is the distance (m) between the two objects,
measured from their centres
G is a constant 6.67 x 10-11 (Nm2kg-2)
Acceleration due to Gravity
Imagine a 1kg mass placed on the Earth’s surface.
Using Newton’s Law of Universal Gravitation, work
out the force F between the Earth and the mass.
F=G
𝑚1 𝑚2
𝑟2
G is 6.67 x 10-11 Nm2kg-2
Mass of the Earth is 5.97 x 1024 kg
Radius of the Earth is 6.37 x 106 m
Acceleration due to Gravity
You should have found that the force is 9.81N.
If we use Newton’s second law f = ma, a formula
linking force (f), mass (m) and acceleration (a),
we can find out the acceleration due to gravity at
the Earth’s surface.
In this case, f = 9.81N and m=1kg
so a = 9.81ms-2
which we often approximate to 9.8ms-2
Weight
Weight is a force and is measured in Newtons (N).
To calculate the weight of an object
we use Newton’s second law f = ma.
Acceleration due to gravity is 9.8ms-2
So, to find the weight of a person whose mass is
60kg we use:
Weight = 60 x 9.8
Weight =588N
Real Weights
Find the weights of the following masses:
• A 1kg bag of flour
• A 7kg baby
• A 15kg suitcase
Is gravity always the same,
everywhere on Earth?
What do you think?
Is the Earth a perfect sphere?
Is gravity always the same,
everywhere on Earth?
The Earth isn’t a perfect sphere;
it’s slightly flatter at the poles,
where the radius is 6.36 x 106 m
(and 6.38 x 106 m at the equator)
F=G
𝑚 1 𝑚2
𝑟2
What weight would a 1kg mass
have at the poles?
Mass of the Earth is 5.97 x 1024 kg
G is 6.67 x 10-11 Nm2kg-2
F=ma
What is the acceleration
due to gravity here?
Is gravity always the same,
everywhere on Earth?
At 8 848m, Mount Everest is the
highest mountain on Earth.
F=G
𝑚 1 𝑚2
𝑟2
Mass of the Earth is 5.97 x 1024 kg
Radius of the Earth is 6.37 x 106 m
G is 6.67 x 10-11 Nm2kg-2
f=ma
What weight would a 1kg mass
have at the top of Everest?
What is the acceleration
due to gravity here?
Is gravity always the same,
everywhere on Earth?
As with many ‘real life’ contexts, things aren’t ‘ideal’
so we use a good approximation that will serve our
purposes reasonably well.
Acceleration due to gravity isn’t the same
everywhere on Earth because it’s not a perfect
sphere.
In calculations we use an average
value of 9.8ms-2 or 9.81ms-2.
A level explanation,
click here
How much would I weigh
on the moon?
Use the data on the next slide to find out how much
you would weigh on the moon, or on some of the
other bodies in our Solar system.
You will still need to use:
F=G
𝑚1 𝑚2
𝑟2
G is 6.67 x 10-11 Nm2kg-2
f=ma
How much would I weigh
on the moon?
Planet
Mercury
Venus
Earth
Mars
Jupiter
Saturn
Uranus
Neptune
Sun
Moon
Radius at equator
(m)
2.44 × 106
6.05 × 106
6.38 × 106
3.39 × 106
7.14 × 107
6.00 × 107
2.61 × 107
2.43 × 107
6.69 × 108
1.74 × 106
Mass (kg)
3.30 × 1023
4.87 × 1024
5.97 × 1024
6.42 × 1023
1.90 × 1027
5.69 × 1026
8.70 × 1025
1.03 × 1026
1.99 × 1030
7.35 × 1022
Walking on the moon
So what do you think it would feel
like to walk on the moon?
On which planet would it feel
most similar to walking on Earth?
Forces in a lift
Think about being in a lift.
If the lift is accelerating upwards, do
you feel heavier or lighter?
What about when it’s accelerating downwards?
What’s the Reaction?
Push your hand against a desk top.
Push it harder.
What do you feel?
Normal Reactions
Assuming you didn’t break the desk(!), you should
have felt that the harder you pushed, the
more the desk seemed to ‘push back’.
You’d have felt a ‘push back’
force if you had pushed against a
wall instead.
This is called the normal reaction force.
(‘normal’ as in ‘at 90° to the surface’, not as in ‘usual’).
This is what we feel.
At rest
When stood still on the ground, we feel a normal
reaction (NR) which, in the case when we are
stood still, is equivalent in to our weight (W).
NR
W
The forces acting are equal and opposite so we
can say the system is ‘in equilibrium’.
Reaction Forces
In fact, it’s a bit more complicated
than it first seems.
Newton’s third law says that for every force there is
an equal and opposite force, sometimes known as an
action-reaction pair.
Beware! Just because the two forces acting are equal
and opposite, doesn’t mean that W and R are an
example of the action-reaction pair mentioned in
Find out more
Newton’s third law!
Lift Off!
When a person is stood in a lift which is
accelerating downwards, the forces acting on the
person are not equal and opposite.
Newton’s second law, f=ma, says that to have an
acceleration, there must be a resultant force, f.
R
a
W
Lift Off!
Supposing a 45kg child is in a lift
which is accelerating at 2ms-2
downwards.
Using Newton’s second law,
f=ma, we have:
W-R = ma
441 - R = 45 x 2
R = 441-90 = 351N
So the child feels lighter.
R
a
W
Weight doesn’t
change
45 x 9.8 = 441N
Lift Off!
Can you work out what the 45kg
child feels in a lift which is
accelerating at 2ms-2 upwards?
R
a
W
A level explanation
If we consider an object of mass 𝑚 𝑘𝑔 a small distance, ℎ,
from the Earth’s surface using Newton’s Law of Universal
Gravitation, with 𝑀 being the mass of the Earth and 𝑅 being
the radius of the Earth:
𝑀𝑚
𝐹=𝐺
𝑅+ℎ
2
𝑀𝑚
=𝐺
𝑅2
ℎ
1+
𝑅
2
ℎ
=𝑔 1+
𝑅
−2
Using the binomial approximation we obtain:
ℎ
1+
𝑅
−2
2ℎ
ℎ
≈1−
+3
𝑅
𝑅
2
ℎ
−4
𝑅
3
+⋯
A level explanation
Since ℎ is very small and especially considering the ratio
we have:
−2
ℎ
2ℎ
1+
≈1−
𝑅
𝑅
ℎ
𝑅
Substituting our expression back into our first equation we
can express the force of gravitational attraction as:
2ℎ
𝐹 ≈𝑔 1−
𝑅
and using values discussed earlier we can see how little the
change is, even on top of Everest!
Back to the activity
More Reactions
Newton’s Law of Universal
Gravitation tells us that two
masses (e.g. a person and the
Earth) attract each other with
the same force.
This is an action-reaction pair.
And in Australia........
W
W’
More Reactions
Newton’s Law of Universal
Gravitation tells us that two
masses (e.g. a person and the
Earth) attract each other with
the same force.
This is an action-reaction pair.
Read on...
W’
W
More Reactions
W
If the person is in contact with
the Earth’s surface then, in
addition to their weight W, they
will experience a Normal
Reaction force, NR.
The person exerts an equal and
opposite Normal Reaction force
on the Earth. The forces on the
Earth are W’ and NR’
NR and NR’ are another
action-reaction pair.
NR
NR’
W’
Action- Reaction Pair
The W and NR are NOT an
action-reaction pair.
The fact that they are equal and
opposite, such as the case
where the person is stood still,
often leads people to think they
are.
W
NR
W’
NR’
Teacher notes: Forces
This issue looks at forces, particularly gravity.
This content will be compulsory in the new Mathematics A level, and is
currently studied within A level Mechanics modules.
In GCSE Mathematics, students don’t need to know about gravity and
other forces, however, in GCSE Science, they do. In GCSE Science,
students also learn about Newton’s laws and the equations of constant
acceleration, so using them in maths may help students connect their
learning in different subjects.
The activities simply require the use of familiar and unfamiliar
equations, so should be accessible to both GCSE and A level students,
and provide opportunities for practising efficient calculator use as well
as consolidating understanding of algebraic rules.
Teacher notes: Forces
» Students should have the opportunity to discuss this
with a partner or in a small group
» Students should sketch or calculate (as appropriate)
What happens?/ What happened
Slide 4
If the items are dropped at the same time, from the same height, they
should hit the ground at the same time.
• There may be some air resistance which will slow down the object
with the larger surface area, but over this distance, the effect should
be negligible.
Slide 5
If the distance is the same, and the time is the same (one sound), and
they both start from rest, then the acceleration must be the same for
both.
What is Gravity?
Slide 7 Note:
Newton’s law of gravitation has been superseded by Einstein’s theory
of general relativity, but we still use Newton’s law as it gives a very
good approximation to the force of attraction in most cases and is much
simpler to use. Where more precision is required, or where the masses
are particularly large or the distances (relatively) close, Einstein’s
theory should be used.
This would make a good discussion point: using something that we
know is wrong, just because it’s easier and is good enough, for now.
Similarly, in mathematics we sometimes overlook things, for now. For
example, KS3 or KS4 students might be told “you can’t find the square
root of a negative number”.
What is Gravity?
Slide 8
It’s worth spending some time understanding the formula and knowing
‘which bits you do first’.
Note: the value of the Gravitational constant G varies according to the
source.
Slide 9
The calculation should result in 9.81N
Slide 12
Flour: 9.8N Baby: 68.6N Suitcase: 147N
Is Gravity always the same?
Slide 13
No, it’s not a perfect sphere. There are mountains and valleys as well
as deep trenches in the ocean. The main shape is not spherical, it’s
slightly flattened at the poles and slightly bulgy around the equator.
Slide 14
At the poles acceleration is 9.84ms-2
At the equator acceleration is 9.78ms-2
Slide 15
Students will need to add Everest’s height to the radius of the Earth to
obtain r.
At the top of Everest, acceleration is 9.79ms-2
How much would I weigh on the moon?
Slide 18
It all depends on the mass of the person. Values given below are for a
1kg mass, so simply multiply by the number of kg required.
Planet
Mercury
Venus
Earth
Mars
Jupiter
Saturn
Uranus
Neptune
Sun
Moon
Radius at
equator (m)
2.44 × 106
6.05 × 106
6.38 × 106
3.39 × 106
7.14 × 107
6.00 × 107
2.61 × 107
2.43 × 107
6.69 × 108
1.74 × 106
Mass (kg)
Weight of 1kg (N)
3.30 × 1023
4.87 × 1024
5.97 × 1024
6.42 × 1023
1.90 × 1027
5.69 × 1026
8.70 × 1025
1.03 × 1026
1.99 × 1030
7.35 × 1022
3.70
8.87
9.78
2.69
24.86
10.54
8.52
11.63
296.57
1.62
Walking on the moon
Slide 19
What’s it like to walk on the moon? Your guess is as good as
anyone’s. On the moon, gravity is about 1/6 of Earth’s gravity. This
means that objects will accelerate to the surface more slowly.
When walking on Earth, gravity helps pulls our feet back down to the
surface, this force would be greatly reduced on the moon, perhaps
resulting in a slower pace and a lighter feeling.
The planet with gravity most similar to ours is Saturn, however, since
this is a Gas Giant, probably consisting of liquids and gases, there is
likely to be no solid surface to walk on!
In terms of gravity, Venus is next closest to our own – and has a solid
surface. However, with a mean temperature of 462°C, humans are
unlikely to walk there in the near future.
Lift Off!
Slide 20
It’s important to emphasise that this is about when a lift is accelerating
(positive or negative acceleration). When the lift is travelling at a
constant velocity, the weight and reaction forces acting on the person
are balanced, as they are when it is at rest.
Slide 24
The ‘find out more’ content is recommended for A level students, but is
equally accessible to others.
Slide 25
The idea of a resultant force is important. In this case the two forces
are in opposite directions, and there is an excess in one direction.
Refer to directed numbers:
-2+2=0 this is balanced, the numbers are ‘equal and opposite’
-2+3=+1 this has a result in a specific direction and is different to
+2+-3=-1
Lift Off!
Slide 27
Accelerating upwards: using Newton’s second law, f=ma, we have:
R-W = ma
R - 441 = 45 x 2
R = 441+90 = 531N
So the child feels heavier.
In the lift activities, action-reaction pairs are mentioned. These are a
very common source of confusion for students and will be explored in
more details in an M4 classroom resource next academic year.
Acknowledgements
https://en.wikipedia.org/wiki/Earth_radius#Polar_radius
https://en.wikipedia.org/wiki/Isaac_Newton
https://en.wikipedia.org/wiki/Galileo_Galilei
Thank you to Simon Clay and Sharon Tripconey for providing the ideas
for these activities.
If you would like to explore these themes further you may be interested
in the Get Set for Mechanics course which will be run by MEI in
preparation for the new A level. See the webpage for details.