# Forces Workbook - EDITABLE Student Workbook (1) ```Forces Equations
Forces
⃑ 𝒏𝒆𝒕 = 𝒎𝒂
⃑
𝑭
⃑ 𝒏𝒆𝒕 = 𝑭
⃑ 𝟏+𝑭
⃑ 𝟐+𝑭
⃑ 𝟑+⋯
𝑭
𝒎
⃑ 𝒈 = 𝒎𝒈
⃑⃑ where 𝒈
⃑⃑ = 𝟗. 𝟖 𝟐 [𝒅𝒐𝒘𝒏]
𝑭
𝒔
⃑𝑭𝑮 =
𝑮𝒎𝟏 𝒎𝟐
𝒓𝟐
𝑭𝑭 = 𝝁𝑭𝑵
⃑𝑭𝒔 = 𝒌∆𝒙
⃑
Trigonometry
Right-Angled Triangles
ℎ𝑦𝑝
𝑜𝑝𝑝
𝜃
Any Triangle
𝑐
𝐵
𝐴
𝑎𝑑𝑗
𝑃𝑦𝑡ℎ𝑎𝑔𝑜𝑟𝑒𝑎𝑛 𝑇ℎ𝑒𝑜𝑟𝑒𝑚
𝑎
𝐶
𝑏
ℎ𝑦𝑝2 = 𝑎𝑑𝑗 2 + 𝑜𝑝𝑝2
𝑆𝑖𝑛𝑒 𝐿𝑎𝑤
sin 𝐴 sin 𝐵 sin 𝐶
=
=
𝑎
𝑏
𝑐
𝑇𝑟𝑖𝑔 𝐼𝑑𝑒𝑛𝑡𝑖𝑡𝑖𝑒𝑠
𝑜𝑝𝑝
sin 𝜃 =
ℎ𝑦𝑝
𝐶𝑜𝑠𝑖𝑛𝑒 𝐿𝑎𝑤
𝑐 2 = 𝑎2 + 𝑏 2 − 2𝑎𝑏 cos 𝐶
𝑎𝑑𝑗
ℎ𝑦𝑝
𝑜𝑝𝑝
tan 𝜃 =
𝑎𝑑𝑗
cos 𝜃 =
&copy; Michelle Brosseau, Mrs. Brosseau’s Binder
Forces Vocabulary
Date:
Name:
Update this list throughout the unit with new Physics vocabulary.
Term
Definition
&copy; Michelle Brosseau, Mrs. Brosseau’s Binder
Forces Preconceptions
Date:
Name:
Answer these questions at the beginning of the Forces unit using the
knowledge you already have. You will revisit these same questions at the
end of the unit.
What is a “force”?
List all the forces you know.
What is “inertia”?
Does an object moving with a
constant velocity require a force
to keep it moving?
How can one lose weight without
exercising?
What are you excited to learn about in this unit?
&copy; Michelle Brosseau, Mrs. Brosseau’s Binder
Newton’s First Law: The Law of Inertia
Date:
Name:
Motion: Aristotle to Galileo
Aristotle, a Greek philosopher and scientist (384-322 BC), uses common
sense to explain physics. He noticed that some objects in motion
maintained their motion without assistance. He called these
“_________________________________”. These included rocks falling off
ledges, liquids running downhill, air rising and flames going upward. His
view was that any solid falls toward the Earth because it seeks to get
closer to its natural resting place. Any motion that was not natural was
“_________________________________,” that is, the object needed to be
pushed or pulled. Aristotelian physics had a hard time explaining many
things, though (sadly) most people still share his views on motion.
Consider this: what happens when you drop two pieces of paper, one is
flat and one is crumpled into a ball?
What happens when you drop two objects of roughly the same shape
but very different masses?
Aristotelian physics could not explain either of these effects. His theory
stated that the heavier object would seek the Earth’s center more
strongly, thus falling faster. We can test the hypothesis that any two
objects dropped from the same height will hit the ground at precisely the
same time using a vacuum (a container that contains no air). This has
been tested, and the objects fall precisely at the same rate. This leads to
an important discovery by Galileo Galilei (1564-1642):
Galileo’s Law of Falling
&copy; Michelle Brosseau, Mrs. Brosseau’s Binder
This means that any two objects – a duck, a velociraptor, a notebook, a
piece of chalk, even an individual atom – will fall together.
Draw a picture here to help you remember Galileo’s Law of Falling.
Astronauts confirmed this on the moon as well. A hammer and a feather
were dropped together from the same height on the moon, and fell at
the same rate, hitting the moon’s surface at the same time.
Cool fact: The falling of an individual atom was tested in 1999, and in
order to do so the researchers had to eliminate any thermal motion the
atom had. This meant cooling the atoms within two-millionths of a
degree of absolute zero. The atoms fell just like stones.
Isn’t that cool?
Galileo made many other contributions to the study of forces, including
recognition of the effects of friction and being one of the first scientists to
use the scientific method. His contributions helped future physicists
develop the 3 laws of Newtonian Physics (to be improved upon by
Quantum Physics in the 1900s).
How Things Move
You’ve heard “inertia is a property of matter” from the opening credits to
Bill Nye the Science Guy. But what exactly is inertia?
Inertia
This leads us to Newton’s* First Law: The Law of Inertia.
&copy; Michelle Brosseau, Mrs. Brosseau’s Binder
Newton’s First Law of Motion, The Law of Inertia
*Though this law is attributed to Sir Isaac Newton, it was actually Rene
Descartes who invented this law.
Consider an object in space (far from the gravitational forces of other
objects in space). If the object was stationary to begin with, then it will
_____________________________________ until an ________________________
____________ acts upon it. Similarly, if the object was moving with a
constant velocity then it will ___________________________________________
until an ____________________________________________ acts upon it.
Draw these situations to help remember Newton’s First Law, The Law of
Inertia:
Stationary object
Object with constant velocity
Example:
Use Newton’s First Law to explain why it is important to wear your
seatbelt in a moving vehicle.
&copy; Michelle Brosseau, Mrs. Brosseau’s Binder
Practice
Q1. Some people believe that more massive objects fall faster than less
massive objects. Describe a demonstration you could perform in your
everyday life to clarify this concept for others.
Q2. Define “inertia” in your own words, then rank the following objects
from lowest to highest inertia:
plane, car, transport truck, motorcycle, toy car
Q3. Suppose you are being chased by a massive bull. Use the concept
of inertia to explain why you could escape by running in a zig-zag
pattern.
Q4. Use Newton’s First Law to explain how headrests help prevent
whiplash in motor vehicle accidents. Well-labelled diagrams will aid
&copy; Michelle Brosseau, Mrs. Brosseau’s Binder
Newton’s Second Law: The Law of Motion
Date:
Name:
The main idea to Newton’s theories is that ______________________________
_______________________________ (as opposed to velocities), and
___________________ is needed to keep an object _______________________
________________________________.
In fact, the acceleration, 𝑎, is proportional to total force, 𝐹𝑛𝑒𝑡 . This was
counter-intuitive to most, but this can easily be demonstrated. Imagine a
mass on a table connected by a string to a mass hanging off the edge
of the table. What happens? What happens when the mass hanging off
the edge of the table is doubled? _____________________________________
We write,
𝑎𝛼𝐹
read, “acceleration is proportional to net force.” This means if one is
increased; the other will increase by the same factor. Also,
1
𝑎𝛼
𝑚
read, “acceleration is inversely proportional to mass.” This means that if
one is increased; the other will decrease by the same factor.
𝑎𝛼
𝐹
𝑚
If we put these two proportionalities together we find an object’s
acceleration is proportional to the force exerted upon it divided by its
mass. If the mass is given in kg and the acceleration in m/s2, the
proportionality can be written as an equality:
The unit of force is appropriately called a newton, abbreviated N, where
N = kg &middot; m/s2. 𝐹𝑛𝑒𝑡 , the net force on an object, is the vector sum of all
forces acting upon the object. (Yes, we still use vectors!)
&copy; Michelle Brosseau, Mrs. Brosseau’s Binder
Newton’s Second Law, The Law of Motion
An object’s acceleration is determined by the net force exerted on it
by its environment and by the object’s mass. The direction of
acceleration is the same as the direction of the net force.
Quantitatively, the acceleration is proportional to the net force divided
by the mass:
Example 1:
A 5.7 kg wooden crate experiences a net force of 6.1 N [West].
Determine the crate’s acceleration.
This example is simple. To tackle more complex problems, we will need
to use a diagram to stay organized.
&copy; Michelle Brosseau, Mrs. Brosseau’s Binder
Free body diagrams
One of our initial steps to solving any problem is to draw a picture. A
free-body diagram, abbreviated FBD, is a picture used to solve problems
involving forces.
Example 2:
A raft is free to move on the surface of the water. There is an upward
buoyant force equal in magnitude to the force of gravity, 1225 N
[down]. There is a force of 150 N [E] from the strong winds and 215 N [E]
from the current. Draw the FBD for the raft, then find 𝐹𝑛𝑒𝑡 and 𝑎.
We begin by drawing a rectangle (or square, or circle) and the object
inside the shape. We also include the mass (in kg) of the object within
the shape.
We then draw all forces acting upon the object using vectors pointing
in the appropriate directions (for most questions this will be up, down,
left and right). We generally choose to use the standard reference
system, where up/North and right/East are positive while left/West and
down/South are negative.
Notes:
Only draw the forces that are ______________________ the object. We
do not include any forces the object is exerting on other objects (like
the raft pushing the water).
Multiple forces in the same direction are drawn next to each other.
If a force is acting __________________ on an object we draw this force
_______________ the shape.
Force vectors always point __________________ the object.
The length of force vectors are relative to their strength, that is
_________________ forces have _______________ vectors.
&copy; Michelle Brosseau, Mrs. Brosseau’s Binder
To find 𝐹𝑛𝑒𝑡 we use the equation:
𝐹𝑛𝑒𝑡 = 𝐹1 + 𝐹2 + 𝐹3 + 𝐹4 + ⋯
That is, the net force is the (vector) sum of all the forces acting on the
object. Consider the horizontal and vertical components separately
&copy; Michelle Brosseau, Mrs. Brosseau’s Binder
More complicated examples require vector addition to find the net force
and acceleration.
Example 3:
A 19-kg sled is being pulled on ice by two siblings, each holding a rope.
The younger sibling pulls with a force of 17 N [E15&deg;N] and the elder
sibling pulls with a force of 21 N [E12&deg;S]. Find the net force and
acceleration of the sled, assuming no friction on the ice.
&copy; Michelle Brosseau, Mrs. Brosseau’s Binder
Practice
Q1. Summarize Newton’s second law in your own words.
Q2. What is a “newton”? What does it measure? Show 1 N as its base SI
units.
Q3. A trio of students push a 65 kg crate. The first student pushes 31 N
[E], the second student pushes 28 N [S] and the third student pushes 39
N [W]. Draw the FBD for the crate.
&copy; Michelle Brosseau, Mrs. Brosseau’s Binder
Q4. For each FBD, find the net force and acceleration.
a)
b)
c)
&copy; Michelle Brosseau, Mrs. Brosseau’s Binder
Q5. Two people push a 2250 kg car, each with a force of 275 N
[forward]. There is a frictional force of 310 N [backwards].
a) Draw the FBD for the car.
b) Find the net force acting on the car.
c) Find the acceleration of the car.
d) Find how far the car has travelled after 10.0 seconds of pushing if
the car starts from rest.
&copy; Michelle Brosseau, Mrs. Brosseau’s Binder
Q6. Two mechanics use pulleys to lift a 105 kg engine out of a car. The
first mechanic pulls with a force of 675 N [U21&deg;R] and the second
mechanic pulls with a force of 712 N [U33&deg;L]. The force of gravity
acting on the engine is 1029 N [D].
a) Draw the FBD for the engine.
b) Determine the net force acting on the engine.
c) Find the acceleration of the engine.
d) Determine how long it will take to lift the engine 1.25 m from rest.
&copy; Michelle Brosseau, Mrs. Brosseau’s Binder
Newton’s Third Law: The Law of Force Pairs
Date:
Name:
Forces always come in pairs. Always.
How do we know?
-slap the tabletop/countertop with your hand
-grab your table, now pull hard on it
-now push hard on the table
When you slap the table, it slaps back – that’s why your hand is stinging.
You exert a force on the table. The table exerts a force back on your
hand because it decelerates your hand (in that it stops your hand)
When you pull on the table, the table pulls you toward it as well.
When you push on the table, the table also pushes you away.
Forces always come in pairs. Always.
Every force is an interaction between two objects – rather than one
object acting on another. Think of slapping the table as an interaction
between your hand and the table – then you should see that each exerts
a force on the other.
If you tap the table lightly, then slap the table hard, can you feel the
difference in the reaction force? This tells us something quantitative
about the force pairs – if one increases the other does too. In fact,
through experimentations we find that the two forces in any force pair
have the same strength. That is – the forces have the same magnitude
(but clearly opposite directions).
Newton saw this too, and summarized it in the Law of Force Pairs:
Newton’s Third Law: The Law of Force Pairs
&copy; Michelle Brosseau, Mrs. Brosseau’s Binder
We can see the interactions between two objects by drawing FBDs for
both objects.
Example:
Draw the FBDs for each object in all of the following situations.
A 100.0 kg hockey player checks a You lean against a wall with a
85.0 kg hockey player with a force force of 25 N.
of 65 N.
You push against the ground as
you walk.
The Earth pulls on the Moon.
&copy; Michelle Brosseau, Mrs. Brosseau’s Binder
Practice
Q1. An apple is pulled towards the ground by the Earth. We can see
the apple accelerate, but we can’t see the Earth accelerate. Does this
break Newton’s third law? Explain.
Q2. Explain these situations using Newton’s third law.
a) A person can move forward by pressing backwards against the
ground.
b) Two skaters, close together, are stationary on the ice. When one
pushes the other they both move away from one another.
c) In space, astronauts can propel themselves in a direction by
throwing an object in the opposite direction.
&copy; Michelle Brosseau, Mrs. Brosseau’s Binder
Q3. A 3500 kg van hits a 2500 kg car with a force of 1480 N [E].
a) What force does the van experience?
b) Calculate the acceleration of both vehicles after the collision.
Q4. If the 3500 kg van and the 2500 kg car each hit each other with a
force of magnitude 1480 N in a head-on collision.
a) Does the net force acting on each vehicle change?
b) Calculate the acceleration of both vehicles after the collision.
&copy; Michelle Brosseau, Mrs. Brosseau’s Binder
Newton’s Laws of Motion Summary
Date:
Name:
Can you fill in the ______________________?
______________________’s Law of Falling
If air ___________________________ is negligible, then any two
_______________________ that are ______________________ together will fall
together, regardless of their ___________________________ and their
___________________________, and regardless of the substances of which
_____________________’s First Law: The Law of ________________________
A body that is subject to no ________________________ forces will stay at
___________________________ if it was at rest to begin with and keep
___________________________ if it was moving to begin with (in which case
its motion will be in a straight ___________________________ at an
unchanging ___________________________).
Or, more simply:
An object that is subject to no external ___________________________ must
___________________________ an unchanging __________________________.
____________________’s Second Law: The Law of ______________________
An object’s ___________________________ is determined by the
___________________________ (2 words) exerted on it by its environment
and by the object’s ___________________________. The direction of
acceleration is the ___________________________ as the direction of the net
force. Quantitatively, the acceleration is proportional to the net force
divided by the mass:
___________________________
____________________’s Third Law: The Law of ___________________________
(2 words)
Every ___________________________ is an interaction between two objects.
Thus forces must come in ___________________________ – if one object
exerts a force on a second object then the second exerts a force on the
first. In fact, the two forces have the same ___________________________
(magnitude) but opposite ___________________________.
&copy; Michelle Brosseau, Mrs. Brosseau’s Binder
Gravity
Date:
Name:
There are four fundamental forces that govern the universe.




Have you ever wondered why the nucleus of an atom stays
together, with all those positively charged protons that repel each
other? The ________________________________________ is responsible
for holding the nuclei of atoms together. It is by far the strongest
force but only acts over short ranges of order 10-15 meters.
The ________________________________________ is the second
strongest force. It causes electric and magnetic effects such as the
repulsion between like electrically charged particles and the
interaction of magnets. It is long-ranged, can be attractive or
repulsive, and acts only between pieces of matter carrying
electrical charge.
The ________________________________________ is responsible for
radioactive decay and neutrino interactions. It has a very short
range and, as its name indicates, it is very weak.
The ________________________________________ is by far the weakest
force, but very long ranged. Furthermore, it is always attractive, and
acts between any two pieces of matter in the Universe.
These four fundamental forces determine the structure of the universe. If
the strength of any of these forces changed the universe would change
dramatically. All four forces are very interesting, but since we’re focusing
on Newtonian Mechanics (as opposed to Quantum Mechanics) let’s
focus on the gravitational force.
Recall the story of Newton and the apple. What caused the apple to fall
on his head? What causes the moon to revolve around the Earth, and
the Earth around the Sun? The answer is the force of ___________________.
Gravity is an ___________________________ force, meaning that the force
always pulls objects closer rather than pushing them away. Also, gravity
is an ________________________________________ force – that is, the apple
didn’t need to be touching the Earth to experience its pull.
Did you know that everything that has mass has a gravitational field?
That means that everything with mass – the Earth, the Sun, an apple, you
and even an electron is attracted to everything else with mass. Have a
look at the person next to you. Did you know that you are attracted to
that person? Gravitationally at least!
&copy; Michelle Brosseau, Mrs. Brosseau’s Binder
In fact, this is how our Sun and planets and satellites were created – bits
of cosmic dust attract each other and combine to create larger cosmic
dust bunnies, which attract even more cosmic dust eventually creating
stars, planets and moons.
Thought Experiment:
Imagine that you and your beloved are the only two things left in the
universe, billions of kilometers apart. Would the two of you ever meet?
What if you both had a velocity that was sending you away from each
other?
On Earth we know that the acceleration due to gravity is about 𝑔 =
𝑚
9.8 2 [𝑑𝑜𝑤𝑛]. Using Newton’s second law, we see that the force of
𝑠
gravity on an object must be:
We use this relationship when we are dealing with objects on or near the
surface of the Earth.
The force of gravity acting on an object is more colloquially called the
object’s weight. Physicists distinguish mass (which is the
_____________________________________ an object has, and is measured in
______) with weight (which is a _____________ and is measured in ______).
If you were to travel to a different planet, one with a different
acceleration due to gravity, your ___________ would remain the same but
your ___________________ would change depending on the
characteristics of that planet.
&copy; Michelle Brosseau, Mrs. Brosseau’s Binder
Example:
A 150-gram apple falls experiences gravity and falls towards the Earth.
Determine the force of gravity acting on the apple.
When an object is in free fall, it does not necessarily accelerate towards
the ground for its entire trip. If air resistance is large enough, the
_________________ force of gravity will _________________ with the
_________________ force of air resistance. When this happens the object
no longer accelerates towards the ground, instead it moves at a
constant velocity called the _________________________________. The
terminal velocity of an object depends on many factors, including the
object’s mass and surface area, and the density of the air.
Example:
Draw the FBD, then find the acceleration of a 65-kg skydiver who
a) jumps out of a plane and experiences 115 N of air resistance.
b) achieves terminal velocity.
&copy; Michelle Brosseau, Mrs. Brosseau’s Binder
Practice
Q1. What does it mean to be an “at-a-distance” force?
Q2. Explain the difference between mass and weight.
Q3. “Force of gravity” and “acceleration due to gravity” are often
confused. Use the example of a small rock and a more massive
cannonball falling from the same height to compare the objects’ force
of gravity and their acceleration due to gravity. Include a diagram.
&copy; Michelle Brosseau, Mrs. Brosseau’s Binder
Q4. Find the force of gravity acting on the following objects:
a) A 1.2 kg rabbit
b) A 55 kg wolf
c) An 800 kg giraffe
d) A 1500 kg hippopotamus
Q5. In 2012, Felix Baumgartner, an Austrian skydiver, ascend into the
stratosphere using a helium balloon. He jumped from a height of 39 km
above Earth’s surface.
a) Do you think that 9.8 m/s2 can be used to determine his
acceleration as he jumped out of the capsule? Why or why not?
b) Draw the FBD for Baumgartner before he pulls his parachute.
c) Baumgartner’s maximum speed was 1357.64 km/h. Draw the FBD
for Baumgartner after he pulls open his parachute.
&copy; Michelle Brosseau, Mrs. Brosseau’s Binder
Universal Law of Gravity
Date:
Name:
The value 𝑔 = 9.8 𝑚⁄𝑠 2 [𝑑𝑜𝑤𝑛] is special. It is a measured value that
applies to objects on or near Earth’s surface. This means, that the value
for 𝑔 could not be used for objects located significantly above or below
the average radius of Earth, or for objects located on other astronomical
objects like planets, dwarf planets and asteroids.
How do we find the force of gravitational attraction between the Earth
and the Moon, between the Earth and the Sun, or between you and your
Physics textbook? Newton to the rescue once again! Newton proved
using Calculus that objects of finite size (objects that can be measured,
even if they are very large) can be considered as particles. That is, even
the largest of objects can be considered as a point, with all its mass at
one point (usually the center of the object).
Using this fact - and the facts that the force of gravitational attraction is
proportional to the masses of both objects in question and inversely
proportional to the distance between them squared – we can develop a
proportionality statement:
But this is not an equality, for the two sides to be equal we need to factor
in the universal gravitational constant, G = 6.67 x 10-11 N∙m2/kg2. (Notice
the universal gravitational constant is capital G, whereas the Earth’s
gravitational constant is lowercase g – think the universe is bigger so it
has the bigger G). Now we have a formula for the force of gravitational
attraction between any two objects:
Where G = 6.67 x 10-11 N∙m2/kg2, m1 is the mass of the first object, m2 is the
mass of the second object and r is the distance between the centers of
the two objects.
&copy; Michelle Brosseau, Mrs. Brosseau’s Binder
Example:
Let’s see how attracted two members of our class are to each other.
This equation works for any two objects in the universe.
Example:
Determine the force of gravity between the Earth and the Sun.
mE = 5.972 x 1024 kg, mS = 1.989 x 1030 kg, r = 1.496 x 1011 m
&copy; Michelle Brosseau, Mrs. Brosseau’s Binder
Practice
Q1. Determine the force of gravitational attraction between a 92 kg
student and a 550 g slice of pizza that are 25 cm apart.
Q2. Neptune’s largest moon, Triton, has an orbital radius of 354 800 km.
Find the force of gravitational attraction between Neptune (mN = 1.024
x 1026 kg) and Triton (mT = 2.14 x 1022 kg).
&copy; Michelle Brosseau, Mrs. Brosseau’s Binder
Q3. Find the weight of a 65 kg person…
a) When they are on the surface of Earth (2 different ways).
b) When they are 36576 m above the surface of Earth (Felix
Baumgartner’s jump altitude).
c) When they are at an altitude 408 km above Earth’s surface (the
same altitude maintained by the International Space Station).
rE = 6371 km, mE = 5.972 x 1024 kg
&copy; Michelle Brosseau, Mrs. Brosseau’s Binder
Q4. Find the of acceleration due to gravity on the surface of Triton.
mT = 2.14 x 1022 kg, rT = 1353 km
Q5. Find the gravitational field strength of Neptune.
mN = 1.024 x 1026 kg, rN = 24622 km
Q6. Explain why different values for the distances are used in Q2, Q4
and Q5.
&copy; Michelle Brosseau, Mrs. Brosseau’s Binder
Physicists’ Weight Loss Plan
Date:
Name:
Choose a person (or object, or pet) and determine its mass in kilograms.
Person/Object
Mass (kg)
Q1. Research the acceleration due to gravity (also called gravitational field strength)
in 3 different cities around the world.
Calculate the person’s weight (in Newtons) for each of the cities.
Acceleration due
City, Country
to gravity (m/s2 or
Person’s Weight (N)
N/kg)
Q2. Research and list at least 2 reasons why the acceleration due to gravity differs
in cities around the world.
Q3. A person wants to lose weight, which area of the world should they move to
and why?
&copy; Michelle Brosseau, Mrs. Brosseau’s Binder
Q4. Research the radius and mass of 3 astronomical objects like planets, dwarf
planets or stars (you need not restrict yourself to our solar system, or even real life Pandora and Tatooine are acceptable!). Then find the person’s weight in Newtons
for each of the three.
Astronomical Object:
Mass:
Find the person’s weight on the surface of this astronomical object.
Astronomical Object:
Mass:
Find the person’s weight on the surface of this astronomical object.
Astronomical Object:
Mass:
Find the person’s weight on the surface of this astronomical object.
&copy; Michelle Brosseau, Mrs. Brosseau’s Binder
Q5. Choose the celestial body from Q4 that gave the person the lowest weight.
Using the radius and mass that you’ve researched, calculate the acceleration due
to gravity (also called gravitational field strength) at the surface of that celestial
body (this is the value that is equivalent to Earth’s 9.8 m/s2).
Determine how many “g”s this is equivalent to by dividing this number by 9.8 m/s2.
Q6. Create a poster, brochure, travel advertisement etc. to show the benefits of the
Physicist’s Weight Loss program.
-Include either the area/city on Earth or the celestial body (or both) that a person
looking to lose the most weight should travel to.
-Include at least 3 benefits/facts about what they will feel with a lower weight.
-Make it very attractive (aesthetically or gravitationally)
&copy; Michelle Brosseau, Mrs. Brosseau’s Binder
The Normal Force
Date:
Name:
Draw the free body diagrams for the following situations:
A 0.225 kg apple is free falling
A 0.225 kg apple is sitting on a
towards Earth.
desk.
Of course, gravity is acting in both situations, but there must be a second
force acting on the apple that is sitting on a desk. How do we know?
Recall Newton’s 2nd Law equation, 𝐹𝑛𝑒𝑡 = 𝑚𝑎. The apple that is sitting on
the desk is _________________________________, so the net force must be
_____________. For the net force to be zero there would have to be an
_______________________________________ acting on the apple that is
keeping it from accelerating. This force is called the
________________________________.
Remember our saying, “all forces come in pairs”, so the force of the
apple pushing down on the tabletop must be met with an equal and
opposite force. The normal force is the __________________________ from
Newton’s Third Law “every action force has an equal and opposite
reaction force.” In this context “normal” does not mean regular or
ordinary – in Math and Physics “normal” means ________________________.
The normal force acts ___________________________________________ to the
surface the object is on.
Draw the normal force, 𝐹𝑁 , acting on the apple in the following situations:
&copy; Michelle Brosseau, Mrs. Brosseau’s Binder
We’ve discussed how the force of gravity can change depending on the
masses and the distance between two objects. Can the normal force
change too?
The Great Compensator
The normal force is the great compensator of forces. Provided the
surface is strong enough to withstand the force applied to it, the normal
force will ________________________ as the force applied increases.
Try This
1. Place your binder/book on the desk.
2. Place your hand on the binder.
3. Apply a large downward force on the binder from your hand.
Did your binder accelerate through the desk? That must mean that the
forces are all _______________________________.
Draw the free body diagrams for both situations.
on the binder.
down hard on the binder.
⃑𝑵
To solve problems using 𝑭
1. Draw a free body diagram and label all forces – expect to label
𝐹𝑁 if there is a surface involved.
2. Create an 𝐹𝑛𝑒𝑡 statement and solve for any unknown variables.
Note: If the surface is on an angle, you will usually need to find the xand y-components of 𝐹𝑁 using Trigonometric Ratios (sine, cosine and
tangent).
&copy; Michelle Brosseau, Mrs. Brosseau’s Binder
Example 1: Normal isn’t always up!
a) An elephant of mass 4700 kg steps onto a platform. What is the normal force
acting on the elephant? Draw the FBD for the elephant.
b) What is the normal force of a magnet being held to the chalkboard with a force
of 1.5 N? Draw the FBD for the magnet.
c) You are helping to install an 11 kg decorative beam to a ceiling. You apply a
force of 170 N upwards to hold the beam in place while you wait for your co-worker
to install the beam. What is the normal force acting on the beam? Draw the FBD
for the beam.
&copy; Michelle Brosseau, Mrs. Brosseau’s Binder
Example 2: Bathroom Scales
Bathroom scale questions are very common when investigating the normal force.
The scale’s reading (called the apparent weight) is always the same as the normal
force acting on the person (force-pairs).
Consider a 75 kg person standing on a scale in an elevator. Hypothesize the
strength of the normal force relative to the force of gravity (greater than, or less
than), then find the normal force if…
a) the elevator is stationary.
b) the elevator accelerates upwards at 0.25 m/s2.
c) the elevator accelerates downwards at 0.6 m/s2.
d) the elevator is moving upwards at a constant velocity.
&copy; Michelle Brosseau, Mrs. Brosseau’s Binder
Practice
Q1. Determine the normal force acting on a 250-gram apple, at rest on
a table. Include an FBD.
Q2. A 1.2 kg textbook rests on top of another 1.2 kg textbook. Draw the
FBDs for each textbook including all values.
Q3. Describe a situation in which the normal force is not directed up.
&copy; Michelle Brosseau, Mrs. Brosseau’s Binder
Q4. An 89-kg person stands on a scale (calibrated in newtons) in an
elevator. Determine the reading on the scale if
a) the elevator is stationary.
b) the elevator accelerates upwards at 0.18 m/s2.
c) the elevator accelerates downwards at 0.75 m/s2.
d) the elevator is moving upwards at a constant velocity.
&copy; Michelle Brosseau, Mrs. Brosseau’s Binder
The Force of Friction
Date:
Name:
The Force of Friction
Consider the apple on the slanted surface. If the
apple is not accelerating, what force is most likely to
prevent the apple from slipping off the surface?
Friction is the Jekyll and Hyde of forces.
We need friction to move, but it also hinders our movement by causing
kinetic (moving) energy to turn into heat and causes wear and tear on
objects.
Friction is believed to be caused by
microscopic welds created by
intermolecular forces. Imagine the
molecules of both substances
“welding” together where they are in
contact to form an attractive force.
To overcome this force (friction) these
welds must be broken. This belief,
along with inertia, also explains why it
is harder to get an object moving
than it is to keep it move.
Friction occurs when two surfaces are touching and is split into two
categories:
1. ____________________________ occurs when an object is not moving
Examples:
2. ____________________________ occurs when an object is moving
Examples:
&copy; Michelle Brosseau, Mrs. Brosseau’s Binder
Criteria for a Frictional Force
1) There exists a “coefficient of friction” – that is, we do not choose to
ignore friction
2) There exists a normal force between two surfaces
3) There is an applied force trying to move that object
If any of these are not met, there is no friction.
Coefficient of Friction, μ
What is a coefficient of friction?
Imagine you have two coins, one on smooth ice and one on rough
plywood. You flick each with an equal force, which is going to move
further before stopping? This is the idea of the coefficient of friction –
different surfaces resist motion differently.
The coefficient of friction is determined by using the ratio between two
forces:
It is the ratio of how much frictional force there is to the normal force
acting on the object. Note how the amount of friction does not depend
on the push or pull applied to the object, instead it is based on the force
of the surface acting on the object – the normal force!
What are the units of the coefficient of friction?
&copy; Michelle Brosseau, Mrs. Brosseau’s Binder
We represent the coefficient of friction by the Greek letter μ, “mu”. There
are two types of coefficients of friction – the coefficient of static friction
(μs) used when the object is stationary, and the coefficient of kinetic
friction (μk) used when an object is moving.
To find the force of friction we re-arrange the previous equation to find:
FF = μsFN for a stationary (static) object
FF = μkFN for a moving (kinetic) object
Coefficients of Friction for Common Material Pairs
Surface A
Surface B
Coefficient of Static
Friction μs
Coefficient of
Kinetic Friction μk
Steel
Steel
Steel
Rubber
Teflon
Wood
Wood
Wood
Human
synovial fluid
Steel
Leather
Copper
Brass
Steel
Concrete
Teflon
Dry snow
Wet snow
Wood
0.53
0.51
0.74
1.10
0.04
0.22
0.14
0.40
0.36
0.44
0.57
1.0
0.04
0.18
0.10
0.20
Cartilage
0.01
0.003
Ice
Rock
0.1
1.0
0.01
0.8
Examine the table.
What range of values does μ fall in?
When would you expect a small
value for μ?
When would you expect a large
value for μ?
&copy; Michelle Brosseau, Mrs. Brosseau’s Binder
For a stationary object to move the force applied must be greater than
the force of static friction.
Example 1: Overcoming Static Friction
A pack of dogs are pulling a wooden sled and rider of combined mass
of 238 kg across dry snow. How much force do the dogs need to apply
to cause the sled to begin to move?
Example 2: Kinetic Friction
You push your 27 kg wooden desk across the wood floor with a
constant velocity. Find the force of friction and applied force.
&copy; Michelle Brosseau, Mrs. Brosseau’s Binder
Sometimes when we perform calculations, the force of static friction is
greater than that of the applied force. Be cautious with the free body
diagrams in these situations. The object will not move until the applied
force is greater than the force of static friction.
Example 3: Will it move?
Two sisters are fighting over the contents of a 48 kg crate. One sister
pulls with a force of 79 N [right], while the other pulls with a force of 51 N
[left]. The coefficient of static friction between the crate and the floor is
0.32. Find the acceleration of the crate.
&copy; Michelle Brosseau, Mrs. Brosseau’s Binder
Practice
Q1. Explain the difference between static and kinetic friction. Which
one has the higher coefficients of friction? In which situations would
you use μs versus μk?
Q2. Draw an FBD for a 3 kg steel block being pulled at a constant
speed on ice. Show all values for the four forces. (Hint: use the
coefficient of friction table in your notes.)
Q3. Explain the concept of coefficient of friction using words that a 5year old would understand.
&copy; Michelle Brosseau, Mrs. Brosseau’s Binder
Q4. Determine the force needed to get a 9 kg piece of rubber moving
on concrete.
Q5. A 2250 kg car has rubber tires and drives on concrete. How much
force must the engine apply to keep the car moving at a constant
speed?
&copy; Michelle Brosseau, Mrs. Brosseau’s Binder
Q6. A child pulls a 5 kg wooden sled on dry snow at a constant velocity
forward. Determine the frictional force if
a) The child pulls the rope at angle 35&deg; above the horizontal.
b) The child pulls the rope at angle 35&deg; below the horizontal.
&copy; Michelle Brosseau, Mrs. Brosseau’s Binder
Mandatory Winter Tires Assignment
Date:
Name:
If you’ve driven down the highway during a harsh winter, surely you have seen the
staggering number of vehicles damaged and abandoned in the ditches. The
number of collisions dramatically increases when the weather turns cold: drivers don’t
change their driving habits and the vehicle is more likely to slide or get stuck in the
presence of ice or snow. The end result is that emergency responders and tow-trucks
are overworked, roads need to be closed down, there is damage to vehicles and
property and passengers and pedestrians are harmed.
The local government is now looking to enforce a law requiring all drivers to put
winter tires on their vehicles. In this assignment you will analyze data on all season
tires vs. winter tires, create a campaign for the mandatory winter tires law and
suggest an appropriate fine for drivers who violate this law.
Create an informative piece that would be distributed to the local citizens explaining
the reasoning behind this new law. This should be a high-quality, professionally
designed piece that will inform and persuade the citizens to start using winter tires.
The format is up to you; brochure, slideshow, commercial, etc.
What to include:
a) A detailed, written explanation into why mandatory winter tires are now
becoming law. This will include a physics-based justification of why winter tires
are superior to all-season tires for use in the cold months of the year.
It may also include statistics that you research into the number of accidents in
Ontario based on weather/road conditions and type of tires, relevant images
(informative or persuasive), cost of repair for damaged vehicles, loss of work
time for common car collision related injuries, etc.
Convince the reader they must invest in winter tires.
b) An analysis of the different stopping distances for one of the vehicle classes
provided (which one might most citizens drive?), across the 8 different tire/road
conditions at 50 km/h, 80 km/h and 100 km/h. That is, calculate how far each
type of vehicle will travel before it stops with different coefficients of friction
between the tires and the road surface. The average citizen requires a visually
impactful message; graph this in an appropriate fashion, or include an
infographic to communicate this data.
This involves a number of calculations. If you are efficient, you will find a way to
automate this calculation (Google Sheets, excel, clever calculator tricks).
you’ll submit them separately from the informative piece.
c) Information on the fine for not using winter tires. The fine for not using winter
tires should be large enough to scare strongly encourage drivers to change
their tires for the winter while balancing the fact that, if charged, the driver still
has to be able to afford the winter tires. Use the following information to
&copy; Michelle Brosseau, Mrs. Brosseau’s Binder
determine and justify how much the fine should be for drivers not using winter
tires during the winter months:
Snow tires vary in price from \$50 - \$450 per tire. Many drivers use inexpensive
rims at about \$50 per wheel for their winter tires and pay \$20 for the tires to be
changed at the shop. If the winter tires are not already on their own rims, the
cost to change the tires increases to about \$90 twice a year (winter tires put on
and taken off). Winter tires generally last 5 years before needing to be
replaced.
The citizens may appreciate a breakdown of how you determined the
appropriate fine. You may include the calculations in your informative piece,
or separately with your calculations to part b.
In addition to the fine amount, you can decide where the funds from this fine
would go (for example, road improvements, police enforcement, etc.),
consequences for not paying the fine (car impounded, jail time, etc.) and
justify that with an explanation.
Data:
Calculate the stopping distances for one of the different classes of vehicles under the
different tire and road conditions at 50 km/h, 80 km/h and 100 km/h.
(1 vehicle x 4 road conditions x 2 types of tires x 3 initial speeds = 24 values)
Vehicle Class
Compact car
Midsize car
Large car
Compact Truck or SUV
Midsize Truck or SUV
Large Truck or SUV
School Bus
Transport Truck
Average
Mass in
kilograms
1350
1590
1985
1575
1940
2460
11000
36000
Coefficient of friction of dry
Coefficient of friction of dry
tires on dry roads, cold day
(-10&deg;C)
Coefficient of friction of tires
on snow
Coefficient of friction of tires
on ice
All Season Tires
Winter Tires
1.0
1.0
0.72
0.89
0.24
0.57
0.05
0.38
Due Date:
&copy; Michelle Brosseau, Mrs. Brosseau’s Binder
Hooke’s Law
Date:
Name:
When you step on a scale or weigh the fish you caught, you are
typically using a device that involves a spring. Whether you
_____________________ a spring (bathroom scale) or
_____________________ a spring (Newton spring scale) the natural
___________________________ of the spring is to bring the spring
back to its natural length (its equilibrium position). Each spring is
different; its _____________________, _____________________ and
_____________________ of the springs affects how much force it
takes to extend or compress the spring. When a spring is being
extended by means of an additional mass, the net force on the
mass will be zero when the spring settles to its new
____________________________________.
The amount of restoring force, 𝐹𝑠 , of a spring is determined by the
extension or compression of the spring (∆𝑥) and its spring constant, 𝑘.
Mathematically,
Hooke’s Law
This equation works for any spring, provided the force applied to the
spring is not too great to cause _____________________ – unravelling of the
spring that is non-repairable.
The Spring Constant, 𝒌
This value is unique to each spring. As discussed before, the material,
thickness and tightness of a spring affects how much it can be stretched
(or compressed). The units of 𝑘 are __________________________________.
What this unit is telling us is that the spring constant is the relationship
between force and stretch (or compression). The value of 𝑘 tells us
______________________________________________________________________.
Therefore, spring with larger values for 𝑘 require ________________________
to cause a stretch (or compression).
Artwork &copy; Glitter Meets Glue Designs
&copy; Michelle Brosseau, Mrs. Brosseau’s Binder
Try These
Complete the table to determine the missing piece of information. Try it
without a calculator.
Spring constant, 𝑘
Stretch/Compression, ∆𝑥
Force applied, ⃑⃑⃑
𝐹
200 N/m
200 N
600 N/m
300 N
25 N/m
100 N
400 N
1m
500 N
0.5 m
700 N/m
0.5 m
250 N/m
2m
One of the many advantages to using Hooke’s Law and that is need not
be restricted to springs. Elastic materials, such as bungee cords and
elastic bands, and flexible materials such as plastics and wood also obey
Hooke’s Law to some extent.
Example 1: Finding the new length
A fish of mass 12.9 kg is attached to a Newton spring scale that has a
spring constant of 665 N/m. Determine the stretch of the spring.
&copy; Michelle Brosseau, Mrs. Brosseau’s Binder
Example 2: Finding the force
A composite hockey stick (k = 9481 N/m) bends 5.0 cm during a slap
shot. How much force did the stick apply to the puck?
Example 3: Finding the spring constant
A 61.2 kg bungee jumper jumps from a bridge. She is tied to a 11.2 m
long (unstretched) bungee cord and falls a total of 32.4 m once her
motion settles. Calculate the spring constant of the bungee cord.
&copy; Michelle Brosseau, Mrs. Brosseau’s Binder
Practice
Q1. A gemstone of mass 1.8 kg compresses a scale’s spring by 2.6 cm.
Determine the spring constant.
Q2. How much would the spring in the previous question compress if a
5.2 kg mass was placed on the scale?
Q3. A Newton spring scale is being calibrated such that 10 N of force
results in an extension of 0.50 cm.
a) Determine the spring constant.
b) Determine the extension of the spring when a mass of 1.36 kg is
suspended from the spring scale.
c) Determine the mass of a suspended object that stretches the
spring by 0.97 cm.
&copy; Michelle Brosseau, Mrs. Brosseau’s Binder
Summary
Define and summarize the characteristics and equations involved with
the following four forces. Include relevant diagrams.
The Force of Gravity
The Normal Force
The Force of Friction
The Spring Force
&copy; Michelle Brosseau, Mrs. Brosseau’s Binder
Systems of Objects
Date:
Name:
An object is in equilibrium when all forces acting upon it are
_______________________. This results in no acceleration for the object
(remember that an object can have all forces balanced and still
maintain its velocity). When two or more objects are involved in a force
problem we call this a ____________________.
If an object is not in equilibrium the forces are _________________________.
This means that the object will accelerate in the direction of the net force
(Newton’s second law).
Systems involving more than one object give you an opportunity to flex
your problem-solving muscles. Several strategies arise in the process of
solving these problems. We will to employ Newton’s second and third
laws frequently.
Most problems can be solved by using the following steps:
1. Read the problem and record known values and unknown
variables.
2. Draw a system diagram that includes all objects.
3. Define the positive and negative directions. It is best practice for
the acceleration to be in the positive direction.
4. Draw an FBD for each object. Use Newton’s third law to identify
any action-reaction force pairs that apply to the FBDs of two
objects.
5. Calculate the values of forces, use horizontal and vertical
components if needed.
6. Use Newton’s second law to solve for missing values. Sometimes,
trigonometry can speed up the solving process.
General Tips:
&copy; Michelle Brosseau, Mrs. Brosseau’s Binder
Example 1: Unbalanced system of two objects
The diagram shows two masses connected via a string
across two pulleys. The masses are initially held in place,
then allowed to move freely. If m1 = 4.7 kg and m2 = 1.8 kg:
a) Which direction will the masses accelerate?
b) Draw the FBD for each mass.
c) Determine the magnitude of acceleration of the
masses.
d) Determine the force of tension in the string.
&copy; Michelle Brosseau, Mrs. Brosseau’s Binder
Example 2: Three objects in equilibrium
As shown in the diagram, three masses are suspended by strings. The system is in
equilibrium, no masses are accelerating. If m2 = 5.0 kg, and the angles are θ = 45&ordm;
and β = 30&ordm;, determine the masses m1 and m3.
&copy; Michelle Brosseau, Mrs. Brosseau’s Binder
Example 3: Perpendicularly accelerating objects
Examine the diagram of two masses connected by a strong, taut string over a
frictionless pulley where m1 = 3.2 kg, m2 = 4.1 kg, μs = 0.35 and μk = 0.28.
a) Show that the two masses will accelerate.
b) Determine the acceleration of the two objects.
c) Determine the force of tension in the string.
&copy; Michelle Brosseau, Mrs. Brosseau’s Binder
The previous problem involved perpendicular forces. How do the problem-solving
strategies change when the forces are not perpendicular?
Examine the system involving two masses connected by a string. One mass is on an
inclined plane, the other mass is suspended over a frictionless pulley.
Draw the free body diagrams for both masses.
If these two masses are allowed to move, we can assume that m 1 would accelerate
up the inclined plane as m2 accelerates downward. Of course, their motion will
depend on the values of 𝑚1 , 𝑚2 , 𝜃, 𝜇𝑠 , and 𝜇𝑘 , but let’s examine how we can simplify
this problem.
&copy; Michelle Brosseau, Mrs. Brosseau’s Binder
Example 4: Objects accelerating on an inclined plane
Examine the diagram of two masses connected by a strong, taut string over a
frictionless pulley where m1 = 7.4 kg is on an inclined plane with θ = 24&deg;, m2 = 5.9 kg,
μs = 0.35 and μk = 0.28.
a) Show that the two masses will accelerate.
b) Determine the acceleration of the two
objects.
c) Determine the force of tension in the string.
&copy; Michelle Brosseau, Mrs. Brosseau’s Binder
Let’s look at this system of accelerating object in general, that is, without any values
associated with each of the variables. We will derive a general equation for the
acceleration of the masses in this system that only involves 𝑚1 , 𝑚2 , 𝑔, 𝜃, and 𝜇𝑘 .
&copy; Michelle Brosseau, Mrs. Brosseau’s Binder
Practice
Q1. Three masses are suspended by strings. If m2 = 12.7 kg, and the angles are θ =
42&ordm; and β = 28&ordm;, determine the masses m1 and m3 that cause this system to be in
equilibrium.
&copy; Michelle Brosseau, Mrs. Brosseau’s Binder
Q2. Three masses are connected by ropes over frictionless pulleys. The masses are
known, m1 = 6.0 kg, m2 = 2.0 kg, and m3 = 1.0 kg. What is the minimum coefficient
of static friction between m2 and the surface to causes all masses to be stationary?
&copy; Michelle Brosseau, Mrs. Brosseau’s Binder
Q3. Examine the diagram of two masses connected by a strong, taut string over a
frictionless pulley where m1 = 13.5 kg is on an inclined plane with θ = 27&deg;, m2 = 12.2
kg, μs = 0.41 and μk = 0.31.
a) Show that the two masses will accelerate.
b) Determine the acceleration of the two
objects.
c) Determine the force of tension in the string.
&copy; Michelle Brosseau, Mrs. Brosseau’s Binder
Q4. Derive a general equation for the acceleration of m2 given that m2 accelerates
upwards.
&copy; Michelle Brosseau, Mrs. Brosseau’s Binder
Q5. Determine the minimum coefficient of static friction such that the masses
remain stationary (assuming that m2 would accelerate downward if not for the
force of friction), written in terms of m1, m2, g and θ.
&copy; Michelle Brosseau, Mrs. Brosseau’s Binder
Q6. Two masses connected via a strong rope across two
pulleys. The masses are initially held in place, then allowed
to move freely. If m1 = 23.7 kg and m2 = 25 kg:
a) Which direction will the masses accelerate?
b) Draw the FBD for each mass.
c) Determine the magnitude of acceleration of the
masses.
d) Determine the force of tension in the string.
&copy; Michelle Brosseau, Mrs. Brosseau’s Binder
Q7. Three masses are connected by ropes over frictionless pulleys. The masses are
known, m1 = 10.2 kg, m2 = 3.1 kg, and m3 = 4.8 kg. Determine the acceleration of
m1 if μs = 0.38 and μk = 0.26.
&copy; Michelle Brosseau, Mrs. Brosseau’s Binder
Forces Preconceptions Revisited
Date:
Name:
After completing the Forces unit, revisit the questions from the beginning
of the unit. What have you learned? Have your responses changed?
What is a “force”?
List all the forces you know.