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Chapter 2
Forces and Vectors
Vector & Scalar Quantities
Vector Quantities
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Vectors are physical quantities that have
both magnitude and direction.
Magnitude = amount and units.
Direction can be stated as up/down,
left/right, N/E/S/W or 35o S of E.
Eg. of vectors: displacement, velocity,
acceleration, force, and momentum.
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Vectors are sometimes represented by a line
and arrow drawn on the line.
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The length of the line represents magnitude
of the vector quantity.
Arrow on the line represents direction.
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When asked to specify a vector quantity,
state both its magnitude (size and units) as
well as its direction.
More about Vectors in Chapter 4!!
Scalar Quantities:
• Scalar quantities are physical
quantities that have only magnitude.
• Scalars do not require direction in
space when specifying them.
• Eg: distance, speed, mass, time,
temperature and energy.
§2.1: Forces
The physical universe is made of objects
(particles) that interact with each other. The
interaction may define or change the
“behavior” (temperature, motion) of the
interacting objects.
Effects of these interactions are explained in
different ways (models) such as force,
momentum exchange, energy, etc.
We will first use force as a means of
understanding some of these interactions.
Force: = Push or pull one object
exerts on another.
• Forces come in pairs, isolated
forces do not exist in physical
interactions.
• Eg. When you push the door,
the door pushes back on you.
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The SI unit of force is the Newton (N).
1N = 1 kg-m/s2.
In the US, force is measured in pounds.
1.00 lb = 4.448 N and 1.00 N = 0.2248 lb.
Robert Hooke (1635 – 1703) found that
when a spring is pulled by a force F, it
extends proportionally by an amount x.
• Hooke’s law: F = -kx
• k = spring constant, indicates how stiff
the spring is.
Measuring Forces
•F = -kx
•Weight = force of
gravity pulling on
objects.
•Extension of the spring
is proportional to the
weight (force). F  x.
•Weight (force) can be
measured using
calibrated spring scale.
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Examples of Forces:
Contact forces – eg. Applied force of
push/pull, force of tension in strings,
friction, normal force, spring force.
Long-range, action-at-a-distance (noncontact) forces. Eg. Gravitational (between
earth-moon), Electric, magnetic.
Weak nuclear force.
Strong nuclear force.
All forces fall under 4 fundamental
categories:
– Gravitational (always attractive).
– Electromagnetic (all contact forces).
– Strong Nuclear (holds protons and
neutrons together.
– Weak Nuclear (occurs in some
forms of radioactivity and
thermonuclear reactions in the sun).
Newton’s Laws of Motion
Sir Isaac Newton (1642-1727)
Newton’s First Law (Also called Law of Inertia)
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An object at rest will remain at rest and an
object in motion will continue to move with
constant speed and direction unless acted
upon by a net force.
Other ways of stating it:
• If no net force acts on an object at rest, it will
remain at rest but if the object is already
moving, it will continue to move without
change in its speed and its direction.
• If the sum of all forces acting on an object is
zero, then its speed and direction will not
change.
 “net force” = vector sum of all forces.
 A net force is needed to make an object
at rest start moving.
 A net force is needed to make a moving
object change direction of motion.
 A net force is needed to stop a moving
object.
 A force is not needed to keep an object
in motion if there is no force opposing its
motion.
§2.4: Net Force and Vector Addition
• Net force = vector sum of all the forces
acting on an object.
• Vectors are added in a special way.
• Co-linear vectors – 2 or more vectors
parallel or antiparallel.
Inertia
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Inertia = resistance to change in motion.
Mass = amount of inertia of an object.
A larger mass has more resistance to change
in its motion than a smaller mass.
An object at rest wants to stay at rest, an
object in motion along a straight line wants to
keep moving that way unless acted on by a
net force. [inertia = resistance to change in
motion] Newton’s Law one  Law of inertia.
Seatbelts are worn because of inertia.
Newton’s first law – closely related to the
reason why seatbelts are worn by motorists.
A force of 15 N is applied to the end
of a spring, and it stretches 9 cm.
How much further will it stretch
if an additional 5.0 N of force is
applied?
(A) 3.0 cm
(B)1.67 cm
(C)10.67 cm
(D)15 cm
(E) 5.0 cm
If the net force acting on a moving
object suddenly becomes zero, the
object will
(A) continue moving but with non zero
acceleration.
(B) stop abruptly.
(C) continue moving at constant
velocity.
(D) slow down gradually.
Velocity (v)
• Velocity (v) in simple terms is speed
and direction. (Better definition later).
• If a nonzero net force (Fnet) is applied to
an object, its velocity will change.
• Thus, if the net force acting on an
object is zero, there will be no change
in its speed, no change in its direction.
• Net force (resultant force) = vector sum
of all the forces acting on an object.
Acceleration (a)
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Change in velocity gives rise to acceleration
(a). If an object moves with changing
velocity, we say the object moves with an
acceleration. Acceleration is rate of change
of velocity. ie a = v/t (“” means “change”).
• Change in velocity could mean
(a) change in speed only - while direction
stays constant.
(b)change in direction only while speed stays
constant.
(c) change in speed and direction of motion
simultaneously. Fnet = ma
Newton’s Second Law:
• The greater the net force, the greater
the acceleration, ie, a  Fnet
• The greater the mass of the object, the
less acceleration, ie, a  1/m.
• The direction of the acceleration is the
same as the direction of the net force.
• Thus, acceleration
a  Fnet/m or a = Fnet/m.
• From Newton’s second law of motion,
we have the relation
Fnet = ma
If a net force of 1 N acts on a 200 gbook, what is the acceleration of the
book?
A box is pushed on a a floor when a
horizontal force of 250 N is applied
against a frictional force of 180 N. If
the box moves with an acceleration of
1.20 m/s2, what is the mass of the box?
Static and Dynamic Equilibrium
If the net force on an object is zero, it could
either:
• Be at rest – static equilibrium.
• Be moving with zero acceleration ie no
change in velocity – constant velocity –
dynamic (or translational) equilibrium.
• If the net force acting on an object is not
zero, the object will move with changing
velocity (acceleration).
Newton’s Third Law of Motion
• In an interaction between two objects,
the forces that each exerts on the other
are equal in magnitude but opposite in
direction.
“To every action, there is an equal an
opposite reaction”
Note that the two equal and opposite forces
are not acting on the same object!!
Action/Reaction Scenarios:
• A person throws a package out of a boat
at rest. Boat starts to move in opposite
direction.
• Ice skater pushes against railing and
moves in opposite direction.
• Rocket exerts strong force expelling
gases. Gases exerts equal force in
opposite direction, propelling the rocket
forward.
Two people pull on a rope in a tag-of-war.
Each pulls with a force of 100 N. The
tension in the rope is
(A)200 N
(B) 100 N
(C) 0 N
(D)Diffeent at different points in the rope
(E)50 N
+ y
+
x
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Force Laws
1. Gravitational Forces:
• Newton’s law of universal gravitation
states that any two objects of masses m1
and m2 separated by a distance r will
exert a gravitational force on each other.
This gravitational force is attractive
force and is directly proportional to the
product of the masses (F  m1m2) and
inversely proportional to r2 (F  1/r2).
• F  m1m2 and F  1/r2 combine to give
F = Gm1m2/r2
• G = Universal Gravitational constant =
6.673 x 10-11 N.m2/kg2
• Objects near the surface of the earth,
gravitational force is called weight, W.
• An object of mass m near the surface
of the earth has weight W = mg
g = acceleration due to gravity = 9.8 m/s2.
r
m1
m2
Gravitational force between m1 and m2 is
F = Gm1m2/r2
m
RE
ME
•Gravitational force between m
and ME is F = GmME/RE2
• This force on m is the weight
of the mass m.
• Weight of m = mg
• mg = GmME/RE2
• Thus g = GME/RE2 = 9.8 m/s2
• Weight of an object of mass m
is W = mg
W = mg = GMEm/RE2 or
g = GME/RE2
• Acceleration due to gravity, g, is directed
downwards, towards the center of the
earth.
• Far away from the surface of the earth,
(r = RE + h), the magnitude of g (and
therefore the weight of an object at that
location), decreases:
g´ = GME/r2 = GME/(RE + h)2
m
RE
m
h
RE
ME
ME
Near the earth’s surface: g = GME/RE2
Far from surface: g´ = GME/r2 = GME/(RE + h)2
Weight (W)
• Is the force of gravity due to the pull of
the earth.
• g = Acceleration due to gravity = 9.8
m/s2.
• Hence for an object of mass m, the
weight is W = mg
• Direction of W is always straight
downward - ie. Toward the center of the
earth.
A man travels to a planet that has the same
mass as the earth, but twice the radius of the
earth. How will his weight on earth (WE)
compared to his weight on this planet (WP)?
(A) WE = WP
(B) WE < WP
(C) WE > WP
(D) It could be any of the above, depending on
his mass.
W = mg
g = GME/RE2
A man travels to a planet that has the same
radius as the earth, but twice the mass of the
earth. How will his weight on earth (WE)
compared to his weight on this planet (WP)?
(A) WE = WP
(B) WE < WP
(C) WE > WP
(D) It could be any of the above, depending on
his mass.
W = mg
g = GME/RE2
2. Spring Force
x
x
• Spring or elastic string
stretched or compressed by
distance x.
• The force that restores the
spring (string) to its original
length is given by the
expression F = -kx [Hooke’s
Law].
• Negative sign is because
direction of F is always
opposite to the direction of x.
3. Normal Force (N)
Consider a book of mass m at rest on a table.
N
mg
By Newton’s law, since the book is at
rest, the net force on it must be zero.
Hence the table must be exerting an
upward force on the book to cancel out
the force of gravity. In this case, N = mg.
[It is not always the case that N = mg!!]
• Normal force is the force on an object when it is
in contact with a surface.
• It is always directed perpendicularly away from
the surface, ie “normally.”
m = 2.0 kg
g = 9.8 m/s2
N
mg
N
mg
(a)
(b)
N
mg
3.0 N
(c)
10.0 N
4. Friction
Friction is a contact force between an
object and a surface, and directed
parallel to the surface. There are two
types of friction:
Static friction and Kinetic friction.
(a) Static Friction: (fs)
• Is the frictional force that exists when there
is no sliding or skidding between an object
and a surface.
• Is the force that keeps an object at rest
against the tendency for it to slide on a
surface.
• Increases to a maximum value fs(max)
when the object starts to slide against the
surface. 0  fs  ffmax
• Maximum static friction ff(max) = sN.
• s = coefficient of static friction.
(b) Kinetic (sliding) Friction:
• Is the frictional force that exists when
an object slides against a surface.
• Is the force that opposes the sliding
movement of an object on a surface.
• fk = kN,
where k is coefficient of kinetic friction.
• Usually, k  s so static friction >
kinetic friction.
N
fk
450 N
W = mg
fk =  k N
W = 750 N
Box sliding at constant velocity. Find:
(a) Mass of the box.
(b) Normal force acting on the box.
(c)Coefficient of kinetic friction for the box-floor.
A box of weight 50 N is at rest on a
floor where s = 0.3.
A rope is attached to the box and
pulled horizontally with tension T =
30 N. Will the box move?
T
50N
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Free Body Diagram
A sketch drawing to help find net force
acting on an isolated (free) body.
Draw the object. May be represented by
just a dot.
Draw all forces acting on the object. The
length of the line and arrows should
represent the forces as closely as possible.
Do not include forces acting on other
objects.
The net force is obtained by performing
vector addition of all the forces drawn.
Identify all forces acting on:
1. A wooden block sliding down an incline
plane.
2. A wooden block sliding up an incline
plane.
3. One of the tires of a car skidding on a flat
road.
4. One of the tires of a car moving
normally on a flat road.
5. A stone in mid air going upward.
6. A stone in mid air coming downward.
Object A is moving with constant
velocity. Object B is at rest. What does
A and B have in common?
(A) Acceleration not zero but constant.
(B) Acceleration is zero.
(C) A non-zero net force acts on them.
(D) Same mass and weight.
The forces acting on a plane are:
Lift L = 14 kN up,
Weight W = -14 kN down,
Thrust T = 0.8 kN east, and
Drag D = 1.2 kN west.
What is the
net force
acting on it?
The moon:
Radius = 1.74 x 106 m
Mass = 7.35 x 1022 kg
What would be the magnitude of
g acting on a mass m placed near
the moon’s surface?
F = Gm1m2/r2, Weight W = mg
G = 6.673 x 10-11 N.m2/kg2
A force of 10 N is applied to the
end of a spring, and it stretches
5 cm. How much further will it
stretch if an additional 5 N of
force is applied?
Hooke’s law: F = -kx