Transcript Forces - Ateneonline

Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Chapter 2: Force
•Forces
•Newton’s First and Third Laws
•Vector Addition
•Gravity
•Contact Forces
•Tension
•Fundamental Forces
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
§2.1 Forces
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.
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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).
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Vectors versus scalars:
A vector is a quantity that has both a magnitude and a
direction. A force is an example of a vector quantity.
A scalar is just a number (no direction). The mass of an
object is an example of a scalar quantity.
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Notation:

Vector: F or F

The magnitude of a vector: F or F or F .
The direction of vector might be “35 south of east”; “20
above the +x-axis”; or….
Scalar: m (not bold face; no arrow)
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§2.2 Net Force
The net force is the vector sum of all the forces acting
on a body.
Fnet  F  F1  F2  F3  
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To graphically represent a vector,
draw a directed line segment.
The length of the line can be used to represent the vector’s
length or magnitude.
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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.
F2
F1
Fnet
For collinear vectors:
F1
F2
Fnet
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§2.3 Newton’s First Law
Newton’s 1st Law (The Law of Inertia):
If no force acts on an object, then its speed and direction of
motion do not change.
Inertia is a measure of an object’s
resistance to changes in its motion.
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If the object is at rest, it remains at rest (speed = 0).
If the object is in motion, it continues to move in a straight
line with the same speed.
No force is required to keep a body in straight line motion
when effects such as friction are negligible.
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An object is in translational equilibrium if the net force on
it is zero.
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
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 in the problem that will provide useful information
for you to solve the given problem.
•Indicate only the forces acting on the body. Label the
forces 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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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
§2.4 Vector Addition
Vector Addition: Place the vectors tip to tail as before. A
vector may be moved any way you please provided that you
do not change its length nor rotate it. The resultant points
from the tail of the first vector to the tip of the second (A+B).
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Example: Vector A has a length of 5.00 meters and points
along the x-axis. Vector B has a length of 3.00 meters and
points 120 from the +x-axis. Compute A+B (=C).
y
B
C
120
A
x
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Example continued:
opp
sin  
hyp
adj
cos 
hyp
sin  opp
tan  

cos adj
y
B
By
60
Bx
sin 60 
By
120
A
x
 B y  B sin 60  3.00m sin 60  2.60 m
B
 Bx
cos60 
 Bx   Bcos60  3.00m cos60  1.50 m
B
and Ax = 5.00 m and Ay = 0.00 m
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Example continued:
C x  Ax  Bx  5.00 m  - 1.50 m  3.50 m
The components of C:
C y  Ay  By  0.00 m  2.60 m  2.60 m
y
The length of C is:
C
Cy = 2.60 m
C  C  Cx  C y
2


x
Cx = 3.50 m
The direction of C is: tan  
Cy
Cx

2
3.50 m 2  2.60 m 2
 4.36 m
2.60 m
 0.7429
3.50 m
  tan 1 0.7429  36.6 From the +x-axis
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
§2.5 Newton’s Third Law
Newton’s 3rd Law:
When 2 bodies interact, the forces on the bodies from each
other are always equal in magnitude and opposite in
direction. Or, forces come in pairs.
Mathematically:
F21  F12.
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Example: Consider a box resting on a table.
F1
(a) If F1 is the force of the Earth
on the box, what is the interaction
partner of this force?
The force of the box on the Earth.
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Example continued:
F2
(b) If F2 is the force of the box on the
table, what is the interaction partner
of this force?
The force of the table on the box.
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External forces:
Any force on a system from a body outside of the system.
F
Pulling a box
across the floor
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Internal forces:
Force between bodies of a system.
Fext
Pulling 2 boxes across the floor
where the two boxes are attached
to each other by a rope.
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
§2.6 Gravity
Gravity is the force between two masses. Gravity is a longrange or field force. No contact is needed between the
bodies. The force of gravity is always attractive!
GM1M 2
F
r2
M1
r is the distance between the two masses
M1 and M2 and G = 6.6710-11 Nm2/kg2.
F12
F21
M2
F21  F12.
r
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Let M1 = mass of the Earth.
 GM E
F  2
 r

M 2

Here F = the force the Earth exerts on mass M2. This is the
force known as weight, w.
 GM E
w   2
 rE

 M 2  gM 2 .


GM E
2
where g 

9
.
8
N/kg

9
.
8
m/s
2
rE
M E  5.98 10 24 kg
rE  6400 km
Near the surface
of the Earth
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F
Note that g 
m
is the gravitational force per unit mass.
This is called the gravitational field
strength. It is often referred to as the
acceleration due to gravity.
What is the direction of g?
What is the direction of w?
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Example: What is the weight of a 100 kg astronaut on the
surface of the Earth (force of the Earth on the astronaut)?
How about in low Earth orbit? This is an orbit about 300 km
above the surface of the Earth.
On Earth:
w  mg  980 N
 GM E 
  890 N
In low Earth orbit: w  mg (ro )  m
 RE  h  
Their weight is reduced by about 10%. The
astronaut is NOT weightless!
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
§2.7 Contact Forces
Contact forces: these forces arise because of an
interaction between the atoms in the surfaces in contact.
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Normal force: this force acts in the direction perpendicular
to the contact surface.
N
Force of the
ground on
the box
w
N
Force of the
ramp on the
box
w
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Example: Consider a box on a table.
y
N
FBD for
box
x
w
Apply
Newton’s
2nd law
F
y
 N w0
So that N  w  mg
This just says the magnitude of the
normal force equals the magnitude
of the weight; they are not Newton’s
third law interaction partners.
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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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Static Friction:
The force of static friction is modeled as
f s  s N .
where s is the coefficient of static friction and N is the
normal force.
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Kinetic Friction:
The force of kinetic friction is modeled as
f k  k N .
where k is the coefficient of kinetic friction and N is the
normal force.
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Example (text problem 2.91): A box full of books rests on a
wooden floor. The normal force the floor exerts on the box is
250 N.
(a) You push horizontally on the box with a force of 120
N, but it refuses to budge. What can you say about the
coefficient of friction between the box and the floor?
y
N
FBD for
box
F
x
fs
w
Apply
Newton’s
2nd Law
(1) Fy  N  w  0
(2) Fx  F  f s  0
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Example continued:
From (2):
F
F  f s   s N   s   0.48
N
This is the minimum value of s, so s > 0.48.
(b) If you must push horizontally on the box with 150 N force
to start it sliding, what is the coefficient of static friction?
Again from (2):
F
F  f s   s N   s   0.60
N
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Example continued:
(c) Once the box is sliding, you only have to push with a
force of 120 N to keep it sliding. What is the coefficient of
kinetic friction?
y
N
FBD for
box
F
x
fk
Apply
Newton’s
2nd Law
(1) Fy  N  w  0
(2) Fx  F  f k  0
w
From 2:
F  f k  k N
F 120 N
k  
 0.48
N 250 N
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
§2.8 Tension
This is the force transmitted through a “rope” from one end
to the other.
An ideal cord has zero mass, does not stretch, and the
tension is the same throughout the cord.
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Example (text problem 2.73): A pulley is hung from the ceiling by a
rope. A block of mass M is suspended by another rope that
passes over the pulley and is attached to the wall. The rope
fastened to the wall makes a right angle with the wall. Neglect the
masses of the rope and the pulley. Find the tension in the rope
from which the pulley hangs and the angle .
y
T
FDB for the
mass M
x
w
Apply Newton’s 2nd
Law to the mass M.
F
y
T w0
T  w  Mg
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Example continued:
Apply Newton’s 2nd Law:
FBD for the pulley:
F
F
y
F
T
x
 F cos   T  0
y
 F sin   T  0
T  F cos  F sin 

x
T
This statement is true
only when  = 45 and
F  2T  2 Mg
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
§2.9 Fundamental Forces
The four fundamental forces of nature are:
•Gravity which is the force between two masses; it is the
weakest of the four.
•Strong Force which helps to bind atomic nuclei together;
it is the strongest of the four.
•Weak Force plays a role in some nuclear reactions.
•Electromagnetic is the force that acts between charged
particles.
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Summary
•Newton’s First and Third Law’s
•Free Body Diagrams
•Adding Vectors
•Contact Forces Versus Long-Range Forces
•Different Forces (friction, gravity, normal, tension)
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
What is the net force acting on the object shown below?
y
15 N
15 N
x
10 N
a. 40 N
b. 0 N
c. 10 N down
d. 10 N up
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
The gravitational field strength of the Moon is about 1/6
that of Earth. If the mass and weight of an astronaut, as
measured on Earth, are m and w respectively, what will
they be on the Moon?
a. m, w
1
b. m, w
6
1
c. m, w
6
1 1
d. m, w
6 6
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