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Chapter 4
The Laws of Motion
Classical Mechanics


Describes the relationship between the
motion of objects in our everyday world
and the forces acting on them
Conditions when Classical Mechanics
does not apply


very tiny objects (< atomic sizes)
objects moving near the speed of light
Forces



Usually think of a force as a push or
pull
Vector quantity
May be a contact force or a field
force


Contact forces result from physical contact
between two objects
Field forces act between disconnected
objects

Also called “action at a distance”
Contact and Field Forces
Fundamental Forces

Types

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Strong nuclear force
Electromagnetic force
Weak nuclear force
Gravity
Characteristics

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All field forces
Listed in order of decreasing strength
Only gravity and electromagnetic in
mechanics
External and Internal
Forces

External force

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Any force that results from the
interaction between the object and its
environment
Internal forces


Forces that originate within the object
itself
They cannot change the object’s
velocity
Net Force

Net force is
the combination of all forces that change an
object’s state of motion.
example: If you pull on a box with 10 N and a friend
pulls oppositely with 5 N, the net force is
5N in the direction you are pulling.
Net Force
Vector quantity


a quantity whose description requires
both magnitude (how much) and
direction (which way)
can be represented by arrows drawn to
scale, called vectors

length of arrow represents magnitude and
arrowhead shows direction
examples: force, velocity, acceleration
The Equilibrium Rule
The equilibrium rule


the vector sum of forces acting on a
non-accelerating object equals zero
in equation form: F = 0
The Equilibrium Rule
example: a string holding up a bag of flour
two forces act on the bag of flour:
–tension force acts upward
–weight acts downward
equal in magnitude and opposite in
direction
when added, cancel to zero
bag of flour remains at rest
Inertia

Is the tendency of an object to
continue in its original motion
Galileo (1564– 1642)
Inertia
Galileo’s Concept of
Inertia
Italian scientist Galileo demolished
Aristotle’s assertions in early 1500s.
Galileo’s discovery
•
•
objects of different weight fall to the ground
at the same time in the absence of air
resistance
a moving object needs no force to keep it
moving in the absence of friction
Galileo’s Concept of
Inertia
Force

is a push or a pull
Inertia


is a property of matter to resist changes
in motion
depends on the amount of matter in an
object (its mass)
Galileo’s Inclinded Plane
The use of inclined planes for Galileo’s experiments
helped him to discover the property called inertia.
Intertia is a property of matter.
Mass—A Measure of
Inertia
Mass
 a measure of the inertia of a material object
 independent of gravity
greater inertia  greater mass
 unit of measurement is the kilogram (kg)
Weight
 the force on an object due to gravity
 scientific unit of force is the Newton (N)

unit is also the pound (lb)
Mass—A Measure of
Inertia
Mass and weight in everyday conversation are
interchangeable.
Mass, however, is different and more fundamental
than weight.
Mass versus weight
•
on Moon and Earth
– weight of an object on Moon is
less than on Earth (1/6th)
– mass of an object is the same
in both locations
Mass—A Measure of
Inertia
One Kilogram Weighs 9.8 Newtons
Relationship between kilograms and
pounds


1 kg = 2.2 lb = 9.8 N at Earth’s surface
1 lb = 4.45 N
Mass



A measure of the resistance of an
object to changes in its motion due
to a force
Scalar quantity
SI units are kg
Sir Isaac Newton

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1642 – 1727
Formulated basic
concepts and laws
of mechanics
Universal
Gravitation
Calculus
Light and optics
Isaac my man
Newton’s First Law

An object moves with a velocity
that is constant in magnitude and
direction, unless acted on by a
nonzero net force

The net force is defined as the vector
sum of all the external forces exerted
on the object
Newton’s Second Law

The acceleration of an object is directly
proportional to the net force acting on it
and inversely proportional to its mass.


F and a are both vectors
Can also be applied three-dimensionally
Units of Force

SI unit of force is a Newton (N)
kg m
1N  1 2
s

US Customary unit of force is a
pound (lb)

1 N = 0.225 lb
Problem 4.1

A 6.0-kg object undergoes an
acceleration of 2.0 m/s2. (a) What
is the magnitude of the resultant
force acting on it? (b) If this same
force is applied to a 4.0-kg object,
what acceleration is produced?
Problem 4.12

The force exerted by the wind on
the sails of a sailboat is 390 N
north. The water exerts a force of
180 N east. If the boat (including
its crew) has a mass of 270 kg,
what are the magnitude and
direction of its acceleration?
Problem 4.19

A 2 000-kg car is slowed down
uniformly from 20.0 m/s to 5.00
m/s in 4.00 s. (a) What average
force acted on the car during that
time, and (b) how far did the car
travel during that time?
Gravitational Force


Mutual force of attraction between
any two objects
Expressed by Newton’s Law of
Universal Gravitation:
m1 m2
Fg  G 2
r
Weight

The magnitude of the gravitational
force acting on an object of mass
m near the Earth’s surface is called
the weight w of the object

w = m g is a special case of Newton’s
Second Law


g is the acceleration due to gravity
g can also be found from the Law
of Universal Gravitation
More about weight

Weight is not an inherent property
of an object


mass is an inherent property
Weight depends upon location
Newton’s Third Law

If object 1 and object 2 interact,
the force exerted by object 1 on
object 2 is equal in magnitude but
opposite in direction to the force
exerted by object 2 on object 1.
 F12  F21

Equivalent to saying a single isolated
force cannot exist
Newton’s Third Law cont.

F12 may be called the
action force and F21
the reaction force


Actually, either force
can be the action or
the reaction force
The action and
reaction forces act
on different objects
Some Action-Reaction
Pairs

n and n '
n is the normal force,
the force the table
exerts on the TV
 n is always
perpendicular to the
surface
 n 'is the reaction – the
TV on the table
 n  n '

More Action-Reaction pairs

Fg and Fg'
 F is the force the
g
Earth exerts on
the object
'
F
 g is the force the
object exerts on
the earth

Fg  Fg'
Forces Acting on an Object



Newton’s Law
uses the forces
acting on an
object
n and Fg are
acting on the
object
'
n ' and Fgare
acting on other
objects
Applications of Newton’s
Laws

Assumptions

Objects behave as particles



can ignore rotational motion (for now)
Masses of strings or ropes are
negligible
Interested only in the forces acting
on the object

can neglect reaction forces
Free Body Diagram
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
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Must identify all the forces acting
on the object of interest
Choose an appropriate coordinate
system
If the free body diagram is
incorrect, the solution will likely be
incorrect
Free Body Diagram,
Example

The force is the
tension acting on the
box


The tension is the same
at all points along the
rope
n and Fg are the
forces exerted by the
earth and the ground
Free Body Diagram, final

Only forces acting directly on the
object are included in the free
body diagram


Reaction forces act on other objects
and so are not included
The reaction forces do not directly
influence the object’s motion
Solving Newton’s Second
Law Problems
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
Read the problem at least once
Draw a picture of the system

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
Identify the object of primary interest
Indicate forces with arrows
Label each force

Use labels that bring to mind the
physical quantity involved
Solving Newton’s Second
Law Problems

Draw a free body diagram

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
Apply Newton’s Second Law

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If additional objects are involved, draw
separate free body diagrams for each object
Choose a convenient coordinate system for
each object
The x- and y-components should be taken
from the vector equation and written
separately
Solve for the unknown(s)
Equilibrium


An object either at rest or moving
with a constant velocity is said to
be in equilibrium
The net force acting on the object
is zero (since the acceleration is
zero)
F  0
Equilibrium cont.

Easier to work with the equation in
terms of its components:
F
x

 0 and
F
y
0
This could be extended to three
dimensions
Equilibrium Example –
Free Body Diagrams
Problem 4.45

(a) What is the resultant
force exerted by the two
cables supporting the
traffic light in Figure
P4.45? (b) What is the
weight of the light?
Problem 4.26

(a) An elevator of mass m moving upward has two forces
acting on it: the upward force of tension in the cable and
the downward force due to gravity. When the elevator is
accelerating upward, which is greater, T or w? (b) When
the elevator is moving at a constant velocity upward,
which is greater, T or w? (c) When the elevator is moving
upward, but the acceleration is downward, which is
greater, T or w? (d) Let the elevator have a mass of 1
500 kg and an upward acceleration of 2.5 m/s2. Find T.
Is your answer consistent with the answer to part (a)?
(e) The elevator of part (d) now moves with a constant
upward velocity of 10 m/s. Find T. Is your answer
consistent with your answer to part (b)? (f) Having
initially moved upward with a constant velocity, the
elevator begins to accelerate downward at 1.50 m/s2.
Find T. Is your answer consistent with your answer to
part (c)?
Inclined Planes


Choose the
coordinate
system with x
along the incline
and y
perpendicular to
the incline
Replace the force
of gravity with its
components
Multiple Objects –
Example




When you have more than one
object, the problem-solving
strategy is applied to each object
Draw free body diagrams for each
object
Apply Newton’s Laws to each
object
Solve the equations
Multiple Objects –
Example, cont.
Problem 4.20

Two packing crates of masses 10.0
kg and 5.00 kg are connected by a
light string that passes over a
frictionless pulley as in Figure
P4.20. The 5.00-kg crate lies on a
smooth incline of angle 40.0°. Find
the acceleration of the 5.00-kg
crate and the tension in the string.
Connected
Objects




Apply Newton’s Laws
separately to each
object
The magnitude of the
acceleration of both
objects will be the
same
The tension is the
same in each diagram
Solve the simultaneous
equations
More About Connected
Objects

Treating the system as one object
allows an alternative method or a
check

Use only external forces



Not the tension – it’s internal
The mass is the mass of the system
Doesn’t tell you anything about
any internal forces
Problem 4.52

Three objects are connected
by light strings as shown in
Figure P4.52. The string
connecting the 4.00-kg object
and the 5.00-kg object passes
over a light frictionless pulley.
Determine (a) the acceleration
of each object and (b) the
tension in the two strings.
Forces of Friction

When an object is in motion on a
surface or through a viscous
medium, there will be a resistance
to the motion


This is due to the interactions
between the object and its
environment
This is resistance is called friction
More About Friction
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Friction is proportional to the normal
force
The force of static friction is generally
greater than the force of kinetic friction
The coefficient of friction (µ) depends
on the surfaces in contact
The direction of the frictional force is
opposite the direction of motion
The coefficients of friction are nearly
independent of the area of contact
Static Friction, ƒs




Static friction acts
to keep the object
from moving
If F increases, so
does ƒs
If F decreases, so
does ƒs
ƒs  µ n
Kinetic Friction, ƒk


The force of
kinetic friction
acts when the
object is in
motion
ƒk = µ n

Variations of the
coefficient with
speed will be
ignored
Block on a Ramp, Example



Axes are rotated as
usual on an incline
The direction of
impending motion
would be down the
plane
Friction acts up the
plane


Opposes the motion
Apply Newton’s Laws
and solve equations
Problem 4.41

Find the acceleration
reached by each of the
two objects shown in
Figure P4.41 if the
coefficient of kinetic
friction between the
7.00-kg object and the
plane is 0.250.
Problem 4.48

(a) What is the minimum force of friction
required to hold the system of Figure P4.48 in
equilibrium? (b) What coefficient of static
friction between the 100-N block and the table
ensures equilibrium? (c) If the coefficient of
kinetic friction between the 100-N block and
the table is 0.250, what hanging weight should
replace the 50.0-N weight to allow the system
to move at a constant speed once it is set in
motion?
Friction of Air
When acceleration of fall is less than
g—non-free fall
•
•
occurs when air resistance is non-negligible
depends on two things: speed and frontal
surface area
Friction of Air
example: A skydiver jumps from plane.




Weight is the only force until air resistance acts.
As falling speed increases, air resistance on diver builds
up, net force is reduced, and acceleration becomes less.
When air resistance equals the diver’s weight, net force
is zero and acceleration terminates.
Diver reaches terminal velocity, then continues the fall
at constant speed.
Friction of Air
Terminal speed

occurs when acceleration terminates
(when air resistance equals weight and
net force is zero)
Terminal velocity

same as terminal speed, with direction
implied or specified
Newton’s Second Law of Motion
CHECK YOUR NEIGHBOR
When a 20-N falling object encounters 5 N of air
resistance, its acceleration of fall is
A.
B.
C.
D.
less than g.
more than g.
g.
terminated.
Newton’s Second Law of Motion
CHECK YOUR ANSWER
When a 20-N falling object encounters 5 N of air
resistance, its acceleration of fall is
A.
B.
C.
D.
less than g.
more than g.
g.
terminated.
Comment:
Acceleration of a nonfree-fall is always less than g.
Acceleration will actually be (20 N – 5 N)/2 kg = 7.5
m/s2.
Newton’s Second Law of Motion
CHECK YOUR NEIGHBOR
If a 50-N person is to fall at terminal speed, the air
resistance needed is
A.
B.
C.
D.
less than 50 N.
50 N.
more than 50 N.
none of the above
Newton’s Second Law of Motion
CHECK YOUR ANSWER
If a 50-N person is to fall at terminal speed, the air
resistance needed is
A.
B.
C.
D.
less than 50 N.
50 N.
more than 50 N.
none of the above
Explanation:
Then, F = 0 and acceleration = 0.
Newton’s Second Law of Motion
CHECK YOUR NEIGHBOR
As the skydiver falls faster and faster through the
air, air resistance
A.
B.
C.
D.
increases.
decreases.
remains the same.
not enough information
Newton’s Second Law of Motion
CHECK YOUR ANSWER
As the skydiver falls faster and faster through the
air, air resistance
A.
B.
C.
D.
increases.
decreases.
remains the same.
not enough information
Newton’s Second Law of Motion
CHECK YOUR NEIGHBOR
As the skydiver continues to fall faster and faster
through the air, net force
A.
B.
C.
D.
increases.
decreases.
remains the same.
not enough information
Newton’s Second Law of Motion
CHECK YOUR ANSWER
As the skydiver continues to fall faster and faster
through the air, net force
A.
B.
C.
D.
increases.
decreases.
remains the same.
not enough information
Newton’s Second Law of Motion
CHECK YOUR NEIGHBOR
As the skydiver continues to fall faster and faster
through the air, her acceleration
A.
B.
C.
D.
increases.
decreases.
remains the same.
not enough information
Newton’s Second Law of Motion
CHECK YOUR ANSWER
As the skydiver continues to fall faster and faster
through the air, her acceleration
A.
B.
C.
D.
increases.
decreases.
remains the same.
not enough information
Newton’s Second Law of Motion
A situation to ponder…
Consider a heavy and light person
jumping together with the samesize parachutes from the same
altitude.
A situation to ponder…
CHECK YOUR NEIGHBOR
Who will reach the ground first?
A.
B.
C.
D.
the light person
the heavy person
both will reach at the same time
not enough information
A situation to ponder…
CHECK YOUR ANSWER
Who will reach the ground first?
A.
B.
C.
D.
the light person
the heavy person
both will reach at the same time
not enough information
Explanation:
The heavier person has a greater terminal velocity. Do
you know why?
Newton’s Second Law of
Motion
Coin and feather fall


feather reaches terminal velocity very quickly
and falls slowly at constant speed, reaching
the bottom after the coin does
coin falls very quickly and air resistance
doesn’t build up to its weight over
short-falling distances, which is why
the coin hits the bottom much sooner
than the falling feather
Air Resistance
How to be a flying squirrel: Jeb Corliss "Grinding the Crack"
Air Resistance
Felix Baumgartner:
Record high sky dive
24 miles high
725 mph

Backup
Problem 4.42

A 2.00-kg block is held in equilibrium
on an incline of angle θ = 60.0° by a
horizontal force applied in the
direction shown in Figure P4.42. If the
coefficient of static friction between
block and incline is μs = 0.300,
determine (a) the minimum value of
and (b) the normal force exerted by
the incline on the block.
Forces on an Elevator



Tension T
Force of gravity
Force of elevator
acceleration
ma
T
mg
T
mg
ma