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Goals for Chapter 4
• To understand the meaning of force in physics
• To view force as a vector and learn how to
combine forces
• To understand the behavior of a body on which
the forces “balance”:
Newton’s First Law of Motion
Goals for Chapter 4
• To learn the relationship between mass,
acceleration, and force:
Newton’s Second Law of Motion
• To relate mass and weight
• To see the effect of action-reaction pairs:
Newton’s Third Law of Motion
• To learn to make free-body diagrams
5-1 Newton's Laws

We will focus on Newton's three laws of motion… but
o
o
Newtonian mechanics is valid for “everyday” situations
It is not valid for speeds which are an appreciable fraction
of the speed of light
o
o
It is not valid for objects on the scale of atomic structure
o
o
Viewed as an approximation of general relativity
Viewed as an approximation of quantum mechanics
And not valid for objects in accelerating frames of
reference!
5-1 Newton's First and Second Laws

Before Newtonian mechanics:
o
o
Some influence (force) was thought necessary to keep a
body moving
The “natural state” of objects was at rest

This seems intuitively reasonable (due to friction)

But envision a frictionless surface
o
Does not slow an object
o
The object would keep moving forever at a constant speed
o
Friction is a force!
What are some properties of a force?
Forces

Characteristics of forces:
o
Unit: N, the newton; 1 N = 1 kg m/s2
o
Forces are vectors

Net force is the vector sum of all forces on an object

Principle of superposition for forces:
o
A net force has the same impact as a single force with
identical magnitude and direction
There are four common types of forces
• #1: The normal force:
When an object pushes on a surface, the surface
pushes back on the object perpendicular to the
surface.
• This is a contact force.
5-2 Some Particular Forces

The normal force:
o
o
If you are standing on a surface, the push back on you from
the surface (due to deformation) is the normal force
Normal means perpendicular
There are four common types of forces
• #2: Friction force:
This force occurs when a surface resists sliding
of an object and is parallel to the surface.
• Friction is a contact force.
There are four common types of forces II
• #3: Tension force:
A pulling force exerted on an object by a rope or
cord.
• This is a contact force.
There are four common types of forces II
• #4: Weight:
The pull of gravity on an object.
• This is a long-range force, not a contact force, and is
also a “field” force.
5-2 Some Particular Forces

The gravitational force:
o
o

A pull that acts on a body, directed toward a second body
Generally we consider situations where the second body is
Earth
In free fall (+y direction chosen as UP, with no drag
from the air):
5-2 Some Particular Forces

In free fall (+y direction chosen as UP, with no drag
from the air):
We can write it as a vector:
5-2 Some Particular Forces
Weight :

The name given to the gravitational force
that one body (like the Earth) exerts on an
object
o
It is a force measured in newtons (N)
o
It is directed downward towards the center
Example To relate weight to mass, consider an apple in
free fall. The only force on the apple is the gravitational
force which results in an acceleration of g. Applying
Newton’s 2nd Law
Fnet = ma
where Fnet = Fg = W and a =g
Example To relate weight to mass, consider an apple in
free fall. The only force on the apple is the gravitational
force which results in an acceleration of g. Applying
Newton’s 2nd Law
Fnet = ma
where Fnet = Fg = W and a =g
Fg = W = mg
Thus,
W = mg
(mass – weight relationship)
Don’t confuse mass and weight!
• Keep in mind that the Newton is a unit of
force, not mass.
• A 2.49 x 104 N Rolls-Royce Phantom traveling
in +x direction makes an emergency stop; the x
component of the net force acting on its -1.83 x
104 N. What is its acceleration?
5-2 Some Particular Forces

Measuring weight:
o
o
o
Use a balance to compare a body to
known masses, find its mass, and
compute its weight
Use a spring scale that measures
weight on a calibrated scale
Weight is not the same as mass: a pan
balance will read the same for different
values of g, a scale will read differently
for different values of g
5-2 Some Particular Forces

Weight must be measured when the body is not
accelerating vertically
o
E.g., in your bathroom, or on a train
o
But not in an elevator
o
Or on DROP ZONE
o
Or in orbit….
5-2 Some Particular Forces
5-2 Some Particular Forces
Example Normal force for a block resting on a horizontal
surface that is:
o
Accelerating vertically at ay:
o
Vertically at rest:
5-2 Some Particular Forces
Example Normal force for a block resting on a horizontal
surface that is:
o
Accelerating vertically at ay:
o
Vertically at rest:
Answer: (a) equal to mg (no acceleration)
(b) greater than mg (with positive acceleration)
5-2 Some Particular Forces

Frictional force or friction:
o
o
Occurs when one object slides or attempts to slide over
another
Directed along the surface, opposite to the direction of
intended motion

Tension force:
o
o
o
o
5-2 Some Particular Forces
A cord (or rope, etc.) is attached to a body and pulled taut
Cord pulls on the body with force T directed along the cord,
AWAY from the body!
The cord is said to be under tension
The tension in the cord is T and is assumed to be the same
EVERYWHERE along the cord!
What are the magnitudes of common forces?
Forces!

A force:
o
Is a “push or pull” acting on an object
o
A net force causes acceleration
o
Newton’s First Law of Motion:
If there is no NET force on an object it will remain in the
same state of motion
Newton’s First Law
• Simply stated:
“An object at rest tends
to stay at rest, an object
in motion tends to stay
in uniform motion.”
Newton’s First Law
• More properly,
“A body acted on by
zero net force moves
with constant velocity
and zero acceleration.”
Newton’s First Law
• In part (a) a net force acts,
causing acceleration.
• In part (b) the net force is
zero, resulting in no
acceleration.
5-1 Newton's First and Second Laws

Newton's first law is true in a non-accelerating frame

Inertial frames:
5-1 Newton's First

Newton's first law is not true in all frames



You go around a corner in a car, and seem to
slide SIDEWAYS!?
Air moving south from the north pole towards
the equator seems to move to the WEST!!!??
You are on a bus moving away from a stop
sign, and your physics books slides
backwards with no one touching it!!!!???
5-1 Newton's First and Second Laws





Newton's first law is not true in accelerating frames
(a): a frictionless puck, pushed from
the north pole, viewed from space
(b): the same situation, viewed from
the ground
Over long distances, the ground is a
non-inertial frame
In (b), a fictitious force would be
needed to explain deflection
When is Newton’s first law valid?
• You are on roller skates
in a stopped BART
car…
• The car starts to
accelerate forwards.
• What happens to you?
• If no net force acts on
you, you maintain a
constant velocity (0!)
When is Newton’s first law valid?
• You are on roller skates
in a moving BART car…
• The car starts to slow accelerate backwards.
• What happens to you?
• If no net force acts on
you, you maintain a
constant velocity
When is Newton’s first law valid?
• If no net force acts on you, you
maintains a constant velocity (a
vector!)
• But as seen in the non-inertial
frame of the accelerating vehicle, it
appears that you are being pushed
to the outside!
• Newton’s first law is valid only in
non-accelerating inertial frames.
When is Newton’s first law valid?
When is Newton’s first law valid?
When is Newton’s first law valid?
When is Newton’s first law valid?
• If NO net force acts on rider, rider maintains a constant
velocity. But as seen in non-inertial frame of the
accelerating vehicle, it appears that rider is being pushed.
• Newton’s first law is valid only in non-accelerating inertial
frames.
Newton’s First Law
•“An object at rest
tends to stay at rest, an
object in motion tends
to stay in uniform
motion.”
Newton’s First Law
•Mathematically,
“A body acted on by
zero net force moves
with constant velocity
and zero
acceleration.”
Newton’s First Law II
• In part (a) net force acts,
causing acceleration.
• In part (b) net force = 0
resulting in no
acceleration.
Drawing force vectors
• Use a vector arrow to indicate the magnitude
and direction of the force.
Superposition of forces
• Several forces acting at a point on an object have
the same effect as their vector sum acting at the
same point.
Decomposing a force into its component vectors
• Choose coordinate system with perpendicular x and y axes.
• Fx and Fy are components of force along axes.
• Use trigonometry to find force components.
5-1 Newton's First and Second Laws

Principle of superposition for forces:
o
o
A net force has the same impact as a single force with
identical magnitude and direction
Restate 1st law even more correctly:
Notation for the vector sum
• Vector sum of all forces on an object is resultant of forces
• The net force.
R=F1+F2 +F3+ = F
Superposition of forces
• Force vectors are added using components:
Rx = F1x + F2x + F3x + …
Ry = F1y + F2y + F3y + …
F1
F2
F3
Superposition of forces
• Force vectors are added using components:
Rx = F1x + F2x + F3x + …
Ry = F1y + F2y + F3y + …
F1
F2
F3
F1
x
Rx
F3
x
Superposition of forces
• Force vectors are added using components:
Rx = F1x + F2x + F3x + …
Ry = F1y + F2y + F3y + …
F1
F2
y
F1
y
F3
Ry
F3
y
Superposition of forces
5-1 Newton's First and Second Laws

Superposition Examples
5-1 Newton's First and Second Laws

Superposition Examples
Answer: (c), (d), (e)
Decomposing a force into its component vectors
• Choose perpendicular x and y axes.
• Fx and Fy are the components of a force along these axes.
• Use trigonometry to find these force components.
Notation for the vector sum—Figure 4.7
• The vector sum of all the forces on an object is called the
resultant of the forces or the net forces.
R= F1+F2 +F3+ = F
Superposition of forces
• Force vectors are most easily added using
components: Rx = F1x + F2x + F3x + … , Ry = F1y + F2y
+ F3y + … . See Example 4.1 (which has three forces).
Newton’s Second Law
•
If the net force on an object is not zero, it causes the object to accelerate.
An object undergoing uniform circular motion
• An object in uniform
circular motion is
accelerated toward
center of circle.
• THEREFORE
Net force on object
must point toward
the center of the
circle.
Force and acceleration
• The acceleration a of an
object is directly proportional
to the net force  Fon the
object.
Mass and acceleration
• The acceleration of an
object is inversely
proportional to the
object’s mass if the net
force remains fixed.
Newton’s second law of motion
• The acceleration of an object is directly
proportional to the net force acting on it, and
inversely proportional to the mass of the object.
F  ma
• The SI unit for force is the newton (N).
1 N = 1 kg·m/s2
Using Newton’s Second Law—Example
5-1 Newton's First and Second Laws

What is mass?
o
o
o
“the mass of a body is the characteristic that relates a force on
the body to the resulting acceleration”
Mass is a measure of a body’s resistance to change in motion
Literally, # of protons, neutrons, & electrons
(regardless of how they are grouped into kinds of elements!)
o
It is not the same as weight, density, size etc.
o
Same force on LARGER mass produces smaller acceleration!
5-1 Newton's
First
and Second
Example Apply identical
8.0 N force
to various
bodies:Laws
o
Mass: 1kg → acceleration: 8 m/s2
o
Mass: 2kg → acceleration: 4 m/s2
o
Mass: 0.5kg → acceleration: 16 m/s2
And if you saw an acceleration from the application of this
force…
o
Acceleration: 2 m/s2
Then you can deduce…
o
Mass = 4 kg
5-1 Newton's First and Second Laws

Summarize these behaviors as:

As an equation, we write:

Identify the body in question, and only include forces
that act on that body!
5-1 Newton's First and Second Laws


As an equation, we write:
Separate applied forces into components along
coordinate system axes
5-1 Newton's First and Second Laws

If the net force on a body is zero:
o
Its acceleration is zero
o
The forces and the body are in equilibrium
o
But there may still be forces! They just must cancel!
Newton’s Second Law
• If the net force on an object is zero, the object will not
accelerate.
Newton’s Second Law
• If the net force on an object is not zero, it causes the
object to accelerate.
Newton’s Second Law
• If the net force on an object is not zero, it causes the
object to accelerate.
An object undergoing uniform circular motion
• An object in uniform
circular motion is
accelerated toward
the center of the
circle.
• So net force on
object must point
toward the center of
the circle.
Force and acceleration
• Magnitude of acceleration of
an object is directly
proportional to the net force
 F on the object.
• | a | = F/m
Mass and acceleration
• Magnitude of acceleration of an
object is
inversely proportional
to object’s mass if net force
remains fixed.
• | a | = F/m
Newton’s second law of motion
• The acceleration of an object is directly
proportional to the net force acting on it, and
inversely proportional to the mass of the object.
F  ma
• The SI unit for force is the newton (N).
1 N = 1 kg·m/s2
Using Newton’s Second Law
• Worker pushes box of mass 40 kg, with constant
force of 20N. What is acceleration?
Using Newton’s Second Law
Using Newton’s Second Law II—Example
• Shove bottle of mass 0.45 kg at initial speed of 2.8
m/s a distance of 1 m before it stops. What is the
magnitude and direction of force on bottle?
Using Newton’s Second Law II—Example
• Shove bottle of mass 0.45 kg at initial speed of 2.8
m/s a distance of 1 m before it stops. What is the
magnitude and direction of force on bottle?
Using Newton’s Second Law II—Example
Using Newton’s Second Law II—Example
• We know:
• Displacement in x = +1.0 m
• Initial x velocity = +2.8 m/s
• Final x velocity = 0 m/s
• THREE THINGS!
Using Newton’s Second Law II—Example
• vf2 = vi2 + 2aDx
+x
• So…
• a = (vf2 - vi2)/2Dx
• a = - 3.9 m/s2
•
NOTE
a
points in the direction of
f
!
Systems of units
• SI: force in Newtons, distance in meters, mass in kg.
• British: force in pounds, distance in feet, & mass in slugs.
• CGS: force in dynes, distance in centimeters, & mass in grams
Mass and weight
• The weight of an object (on the earth) is the
gravitational force that the earth exerts on it.
• The weight W of an object of mass m is
W = mg
• The value of g depends on altitude.
• On other planets, g will have an entirely
different value than on the earth.
A euro in free fall
Don’t confuse mass and weight!
• Keep in mind that
the newton is a
unit of force, not
mass.
Newton’s Third Law
• If you exert a force on a body, the body always
exerts a force (the “reaction”) back upon you.
• A force and its reaction
force have the same
magnitude but opposite
directions.
• These forces act on
different bodies.

Objects interact when they push or pull on each other:

We can write this law as a scalar or vector relation:

We call these two forces a third-law force pair

Any time any two objects interact, there is a third-law
force pair


We call these two forces a third-law force pair
Any time any two objects interact, there is a third-law
force pair
Applying Newton’s Third Law: Objects at rest
• An apple rests on a table. Identify the forces that act
on it and the action-reaction pairs.
Applying Newton’s Third Law: Objects at rest
• An apple rests on a table. Identify the forces that act
on it and the action-reaction pairs.
Applying Newton’s Third Law: Objects at rest
• An apple rests on a table. Identify the forces that act
on it and the action-reaction pairs.
Applying Newton’s Third Law: Objects at rest
• An apple rests on a table. Identify the forces that act
on it and the action-reaction pairs.
Applying Newton’s Third Law: Objects in motion
• A person pulls on a block across the floor. Identify
the action-reaction pairs.
Applying Newton’s Third Law: Objects in motion
• A person pulls on a block across the floor. Identify
the action-reaction pairs.
Applying Newton’s Third Law: Objects in motion
• A person pulls on a block across the floor. Identify
the action-reaction pairs.
Applying Newton’s Third Law: Objects in motion
• A person pulls on a block across the floor. Identify
the action-reaction pairs.
A paradox?
• If an object pulls back on you just as hard as you
pull on it, how can it ever accelerate?
5-3 Applying Newton's Laws
5-3 Applying Newton's Laws

Third-law force pairs:
5-3 Applying Newton's Laws
5-3 Applying Newton's Laws
Answer: (a) they increase
(b) yes
(c) they begin to decrease slowly (the gravitational force
of Earth decreases with height—negligible in an
actual elevator)
(d) yes
A paradox?
• If an object pulls back on you just as hard as you
pull on it, how can it ever accelerate?
Free-body diagrams
• A free-body diagram is a sketch showing all the forces acting
on an object.





To solve problems with forces, we
often draw a free body diagram
The only body shown is the one
we are solving for
Forces are drawn as vector
arrows with their tails on the body
Coordinate system shown
Acceleration is NEVER part of a
free body diagram – only forces
on a body are present.
Free Body Diagrams
Free Body Diagrams
Free Body Diagrams
Free Body Diagrams
Answer: F3
= 2 N to the left in
both cases
5-2 Some Particular Forces
5-2 Some Particular Forces


Real world – Tension will distort a cord/cable/string!
They stretch!
Ideal world - A massless and unstretchable cord exists
only as a connection between two bodies
o
o
o
It pulls on both with the same force, T
True even if the bodies and cord are accelerating, and even
if the cord runs around a massless, frictionless pulley
These are useful simplifying assumptions
5-2 Some Particular Forces
5-2 Some Particular Forces
Answer: (a) equal to 75 N
(b) greater than 75 N (c) less than 75 N
5-3 Applying Newton's Laws
Sample Problem A block of mass M = 3.3 kg, connected by a
cord and pulley to a hanging block of mass m = 2.1 kg, slides
across a frictionless surface
5-3 Applying Newton's Laws





Draw the forces involved
Treat the string as unstretchable, the pulley as
massless and frictionless, and each block as a particle
Draw a free-body diagram
for each mass
Apply Newton's 2nd law
(F = ma) to each block →
2 simultaneous eqs.
Eliminate unknowns (T)
that are the same, and
solve for the acceleration
5-3 Applying Newton's Laws

For the sliding block:

For the hanging block:

Note the direction of “a” for the hanging block!!!

Combining we get:
5-3 Applying Newton's Laws

For the sliding block:

For the hanging block:

Combining we get:

Plugging in we find a = 3.8 m/s2 and T = 13 N

Does this make sense? Check dimensions are correct,
check that a < g, check that T < mg (otherwise
acceleration would be upward), check limiting cases
(e.g., g = 0, M = 0, m = ∞)