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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