forces lecture
TRANSCRIPT
7/31/2019 Forces Lecture
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1Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Forces &
VectorsLecture
Outline
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Forces and Vectors
•Forces
•Hooke’s Law
•Weight
•Contact Forces (normal, friction, tension)
• Apparent Weight
• Air Resistance (drag force)
•Fundamental Forces
•Vector components
•Vector addition
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The quantities in this
column are based on an
agreed upon standard.
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Isaac Newton was the first to discover that the laws that
govern motions on the Earth also applied to celestial bodies.
Over the next few chapters we will study how bodies interact
with one another.
§4.1 Forces
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Simply, a force is a “push” or “pull” on an object.
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How can a force be measured? One way is with a spring
scale.
By hanging masses on a
spring we find that the
spring stretch∝applied
force.
The units of force are Newtons (N).
LawsHooke'calledisThis stretched spring
stretched spring
kx F
x F
=
∝
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The force due to a spring is equal to the amount it is
stretched times the spring constant for that particular
spring. It should be noted here that the direction of the
force is always in the opposite direction to the stretch.
The weight of the
hanging mass is equal to
the mass of the object
itself times a gravitational
factor g (g=9.8 m/s2)
mg F W gravity =≡
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The net force is the vector sum of all the forces acting on
a body.
+++=Σ= 321net FFFFF
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Free Body Diagrams:
•Must be drawn for problems when forces are involved.
•Must be large so that they are readable.
•Draw an idealization of the body in question (a dot, a box,
…). You will need one free body diagram for each body inthe problem that will provide useful information for you to
solve the given problem.
•Indicate only the forces acting on the body. Label theforces appropriately. Do not include the forces that this
body exerts on any other body.
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Free Body Diagrams (continued):
• A coordinate system is a must.
•Do not include fictitious forces. Remember that ma is itself
not a force!
•You may indicate the direction of the body’s acceleration or
direction of motion if you wish, but it must be done well off to
the side of the free body diagram.
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An object is in translational equilibrium if the net force on it
is zero.
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If a = 0, then ΣF = 0. This body can have:
Speed = 0 which is called static equilibrium, or
Speed ≠ 0, but constant, which is called dynamicequilibrium.
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§4.6 Contact Forces
Contact forces: these forces arise because of an interactionbetween the atoms in the surfaces in contact.
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Normal force:
A surface needs to exert a force on an object in order to
stay a solid surface.
In mathematics the definition of “normal” is the vector
that is perpendicular to a surface.
The normal force is the force directed along the normal
direction needed to keep a surface intact.
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Normal force: this force acts in the direction perpendicular to
the contact surface.
Normal force
of the ramp
on the box
N
w
Normal force of the
ground on the boxN
w
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Example: Consider a box on a table.
FBD for
box
mg w N
w N F y
==
=−=∑thatSo
0
This just says the magnitude of thenormal force equals the magnitude
of the weight; they are not Newton’s
third law interaction partners.
Apply
Newton’s
2nd law
N
w
x
y
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Friction: a contact force parallel to the contact surfaces.
Static friction acts to prevent objects from sliding.
Kinetic friction acts to make sliding objects slow down.
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Kinetic friction: the friction experienced by surfaces sliding
against one another
The kinetic frictional force depends on the normal force:
The constant is called the coefficient of kinetic friction.
N k k F f µ =
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The kinetic frictional force is also independent of the relative
speed of the surfaces, and of their area of contact.
g F
N F
k
f
g F 2
N F 2
k f 2
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The static frictional force keeps an object from starting to move
when a force is applied. The static frictional force has a
maximum value, but may take on any value from zero to themaximum,
depending on what is needed to keep the sum of forces zero.
N s s Ff µ≤
s f k f
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§4.7 Tension
An ideal cord has zero mass, does not stretch, and the
tension is the same throughout the cord.
This is the force transmitted through a “rope” from one end
to the other.
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§4.11 Air Resistance
A stone is dropped from the edge of a cliff; if air resistance
cannot be ignored, the FBD for the stone is:
Apply Newton’s Second Law
maw F F d y =−=∑x
y
w
Fd
Where Fd is the magnitude of the drag
force on the stone. This force is
directed opposite the object’s velocity
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Force of fluid resistance depends on the
area of drag and the velocity. Different
fluids behave differently at different
speeds, so we will only consider the
simplest cases here.
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ndrag bv f =
Whereb is a factor of area and shape (different shapes have differing drag
coefficients) andn depends on the properties of a fluid (usually 2 for air
resistance and 1 for water or oil)
Like the other frictional forces the drag force always acts against the direction
of motion.
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©2008 by W.H.Freeman and
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Example: A paratrooper with a fully loaded pack has a
mass of 120 kg. The force due to air resistance has a
magnitude of Fd = bv2 where b = 0.14 N s2/m2.
(a) For what speed (v) is the wieght of the paratrooper
balanced by the air drag?
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§9.6 Archimedes’ Principle
An FBD for an object floating
submerged in a fluid.
The total force on the block due to the fluid is called thebuoyant force.
12
12
where FF
FFF
>
−= B
wF2
F1
x
y
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gV g m F F fluid fluid g B ρ === ,
Archimedes’ Principle: A fluid exerts an upward buoyant
force on a submerged object equal in magnitude to the
weight of the volume of fluid displaced by the object.
31000kg/miswater of DensityThe
fluidaof densityor per volumeMass≡=
V
m ρ
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Example (text problem 9.28): A flat-bottomed barge loaded
with coal has a mass of 3.0× 105 kg. The barge is 20.0 m long
and 10.0 m wide. It floats in fresh water. What is the depth of
the barge below the waterline?
x
y
w
FB
FBD for
the barge
The weight of the barge is balanced by
the buoyant force:
( )
( ) bw
bww
bwww
B
B
m Ad
mV
g m g V g m
w F
w F F
=
=
==
=
=−=∑
ρ
ρ
ρ
0
( )( )m5.1
m10.0*m0.20kg/m1000
kg100.33
5
=×
== A
md
w
b
ρ
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§4.12 Fundamental Forces
Fundamental Interactions of Nature
1. The gravitational force
• The only one of these we will be looking at in this course
• The weakest of the forces listed here
• Only noticeable when a very large mass is involved
• Acts over a long distance
1. The electromagnetic interaction
• Responsible for holding molecules together
• At least 1023 times stronger than gravity
• Will be discussed at length in 2nd semester of this class
1. The weak interaction
• Responsible for some types of radioactive decay
1. The strong interaction
• Holds the nucleus of an atom together.
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§3.1 Graphical Addition and
Subtraction of Vectors
A vector is a quantity that has both a magnitude and a
direction. Position is an example of a vector quantity.
A scalar is a quantity with no direction. The mass of an
object is an example of a scalar quantity.
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Notation:
Vector: FF
or
The magnitude of a vector: .or or FF
F
Scalar: m (not bold face; no arrow)
The direction of vector might be “35°south of east”; “20° above the +x-axis”; or….
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§3.2 Vector Addition and
Subtraction Using Components
Vectors may be moved any way you please (to place them tipto tail) provided that you do not change their length nor rotate
them.
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The Components of a Vector
Understanding direction:
•North, South, East, & West are understood to be directions
•When graphing x & y are often separated like this (x,y)
•In physics the ^ hat symbol over any vector is used to signify
in the direction of that vector and is called a unit vector
•Primarily we use to show “in the x direction” &
“in the y direction”
•Unit vectors can be multiplied by any scalar quantity to showan amount along that direction.
y x ˆandˆ
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The Components of a Vector
x
y
A
x A
y A
θ
θ cos A=
θ sin A=
A
A A
vector theof lengthThe
=
directionshowtoˆ&ˆrsunit vecto by
multiplied bemusthenceandscalarsare&that Note
ˆˆ
componentssit'of sumtheaswriten becanvector The
y x
A A y A x A A
A
y x
y x+=
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The Components of a Vector
x
y
A
x A
y A
φ
φ cos A=
φ sin A=
A
A A
vector theof lengthThe
=
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The Components of a Vector
Signs of vector components:
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To add vectors graphically they must be placed “tip to tail”.
The result (F1 + F2) points from the tail of the first vector to
the tip of the second vector.
F1
F2
R
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Think of vector subtraction A − B as A+(−B), where the vector
−B has the same magnitude as B but points in the opposite
direction.
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x
y
A
x A
y A
y B
x B
B
C
x x x B AC +=
y y y B AC +=
Adding and Subtracting Vectors
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