PHYS 1443 – Section 002
Lecture #9
Wednesday, Oct. 1, 2008
Dr. Jaehoon Yu
•
•
Free Body Diagram
Application of Newton’s Laws
–
•
Motion without friction
Forces of Friction
–
Wednesday, Oct. 1, 2008
Motion with friction
PHYS 1443-002, Fall 2008
Dr. Jaehoon Yu
1
Reminder: Special Project for Extra Credit
• Show that the trajectory of a projectile motion
is a parabola!!
– 20 points
– Due: Monday, Oct. 6
– You MUST show full details of
computations to obtain any credit
• Beyond what was covered in the lecture note!!
Wednesday, Oct. 1, 2008
2
Some Basic Information
When Newton’s laws are applied, external forces are only of interest!!
Why?
Because, as described in Newton’s first law, an object will keep its
current motion unless non-zero net external force is applied.
Normal Force, n:
Reaction force that reacts to action forces due to
the surface structure of an object. Its direction is
perpendicular to the surface.
Tension, T:
The reactionary force by a stringy object
against an external force exerted on it.
Free-body diagram
Wednesday, Oct. 1, 2008
A graphical tool which is a diagram of external
forces on an object and is extremely useful analyzing
forces and motion!! Drawn only on an object.
3
Categories of Forces
• Fundamental Forces: Truly unique forces that
cannot be derived from any other forces
– Total of three fundamental forces
• Gravitational Force
• Electro-Weak Force
• Strong Nuclear Force
• Non-fundamental forces: Forces that can be
derived from fundamental forces
– Friction
– Tension in a rope
– Normal or support forces
Wednesday, Oct. 1, 2008
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Free Body Diagrams and Solving Problems
•

1.
2.
3.
4.
5.
6.

Free-body diagram: A diagram of vector forces acting on an object
A great tool to solve a problem using forces or using dynamics
Select a point on an object in the problem
Identify all the forces acting only on the selected object
Define a reference frame with positive and negative axes specified
Draw arrows to represent the force vectors on the selected point
Write down net force vector equation
Write down the forces in components to solve the problems
No matter which one we choose to draw the diagram on, the results should be the same,
as long as they are from the same motion
FN
M
FN
Which one would you like to select to draw FBD?
What do you think are the forces acting on this object?
Gravitational force
FG  M g
Gravitational force
Me
m
The force pulling the elevator (Tension)
What about the box in the elevator?
Wednesday, Oct. 1, 2008
FG  M g
the force supporting the object exerted by the floor
Which one would you like to select to draw FBD?
What do you think are the forces acting on this elevator?
FT
FN
F GB  mg
FG  M g
Gravitational
force
Normal
force
FT
FG  M g
FN
F5BG  m g
Applications of Newton’s Laws
Suppose you are pulling a box on frictionless ice, using a rope.
M
What are the forces being
exerted on the box?
T
Gravitational force: Fg
n= -Fg
Free-body
diagram
n= -Fg
T
Fg=Mg
Normal force: n
T
Fg=Mg
Tension force: T
Total force:
F=Fg+n+T=T
If T is a constant
force, ax, is constant
Wednesday, Oct. 1, 2008
 Fx  T  Ma x
F
y
ax 
  Fg  n  Ma y  0
T
M
ay  0
v xf  vxi  axt  v xi  
T 
t
M 
1 T  2
x

x

v
t


t
x  f i xi
2M 
What happened to the motion in y-direction?
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Example for Using Newton’s Laws
A traffic light weighing 125 N hangs from a cable tied to two other cables
fastened to a support. The upper cables make angles of 37.0o and 53.0o with
the horizontal. Find the tension in the three cables.
37o
y
53o
T1
Free-body
Diagram
T2
37o
53o
T3
r r r
r
r
T

T

T

ma
0
F 1 2 3
i 3
x-comp. of
Tix  0
net force Fx  
i 1
y-comp. of
net force Fy 
i 3
T
i 1
iy
0
Wednesday, Oct. 1, 2008
 
x
Newton’s 2nd law
 
 T1 cos 37  T2 cos 53  0 T1 
 
 
T sin 53   0.754  sin 37   1.25T
 T
cos  37 
cos 53o
o
2

0.754T2
T1 sin 37o  T2 sin 53o  mg  0

2
T2  100 N ;

2
 125N
T1  0.754T2  75.4 N
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Example w/o Friction
A crate of mass M is placed on a frictionless inclined plane of angle q.
a) Determine the acceleration of the crate after it is released.
y
r
ur ur
r
F F g  n ma
n
Fx  Ma x  Fgx  Mg sin q
y
n
x
q
q
Fg
Free-body
Diagram
Supposed the crate was released at the
top of the incline, and the length of the
incline is d. How long does it take for the
crate to reach the bottom and what is its
speed at the bottom?
ax  g sin q
x
F= -Mg F  Ma  n  F  n  mg cos q  0
y
y
gy
1 2 1
v
t

a x t  g sin q t 2  t 
d  ix
2
2
v xf  vix  axt g sin q
2d
g sin q
2d
 2dg sin q
g sin q
 vxf  2dg sin q
Wednesday, Oct. 1, 2008
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Forces of Friction
Resistive force exerted on a moving object due to viscosity or other types
frictional property of the medium in or surface on which the object moves.
These forces are either proportional to the velocity or the normal force.
Force of static friction, fs: The resistive force exerted on the object until
just before the beginning of its movement
Empirical
Formula
ur
r
f s  s n
What does this
formula tell you?
Frictional force increases till it
reaches the limit!!
Beyond the limit, the object moves, and there is NO MORE static
friction but kinetic friction takes it over.
Force of kinetic friction, fk
ur
r
f k  k n
The resistive force exerted on the object
during its movement
Which direction does kinetic friction apply?
Wednesday, Oct. 1, 2008
Opposite to the motion!
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Example w/ Friction
Suppose a block is placed on a rough surface inclined relative to the horizontal. The
inclination angle is increased till the block starts to move. Show that by measuring
this critical angle, qc, one can determine coefficient of static friction, s.
y
n
Free-body
Diagram
Fg
n
fs=kn
x
F= -Mg
q
q
Net force
r ur
r ur
ur
F  M a  Fg n f s
x comp.
Fx  Fgx  f s  Mg sin q  f s  0
f s   s n  Mg sin q c
y comp.
Fy  Ma y  n  Fgy  n  Mg cosqc  0
n Fgy  Mg cosq c
Mg sin q c
Mg sin q c
 tan q c
s 

Mg cos q c
n
Wednesday, Oct. 1, 2008
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