For next lecture:
Forces
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Review type I, II, and III levers (including the
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concepts of “resistance arm” and “effort arm”)
be prepared to discuss biomechanical
examples of the various types in class
Ozkaya and Nordin, Chapter 1 (p. 7-12),
Chapter 2 (p. 19-28)
Outline
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units
scalars and vectors
transmissibility of force
gravitational forces
frictional forces
parallelogram law for addition of forces
coordinate systems and components of force
Units
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base units:
length l: meters (m)
mass m: kilograms (kg)
time t: seconds (s)
temperature, degrees Kelvin (K)
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supplementary unit:
plane angle !: radian (rad)
Derived units
Unit conversions
Multiply:
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lb by 4.45 to get N
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inches by 0.0254 to get m
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in·lb by ?? to get N·m?
1 [in·lb]·4.45 [N/lb] ·0.0254 [m/in] = 0.113 N·m
Vectors and Scalars
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Scalars: variables defined by magnitude only
(e.g., volume, temperature)
Vectors: variables defined by magnitude and
direction (e.g., force, velocity)
v = 10 m s-1
30°
NOTE: We will designate
vectors with arrows in
diagrams, and with boldface
(r) or an arrow over the
symbol (r ) in text. We will
use italics (r) when referring
to only the magnitude of a
vector quantity.
Forces are vectors
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force: push or pull that tends
to cause motion of a body, or
change the shape of a body
F
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characteristics of a force:
! point of application
! magnitude
! direction (sense and line
of action)
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Transmissibility Of Force
Same equilibrium conditions, but
different internal forces?
The force acting on a rigid body can
“slide” along it’s line of action, without
changing the conditions of equilibrium or
motion.
r
r
Fr = W
r
W = mg
=
r
W = mg
r
r
Fr = W
Gravitational forces
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gravitational force,
or weight, is an
attractive force of
magnitude mg that
the earth exerts on a
body of mass m
g = gravitational
constant = 9.81 m/s2
example of a noncontact force
Measuring force
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r
W = mg
r
r
Fr = W
Force cannot be measured, only derived
from other measures (e.g., deflection of a
spring).
deflection
(units of m)
F
k
"x
F = k"x
spring constant
(units of N/m)
Resolution of force into components
Adding forces by summing components
• i and j are unit vectors
in the x and y direction,
respectively.
• Fx and Fy are
magnitudes
• Fx and Fy are vectors in
x and y direction,
respectively
Force Addition (Parallelogram Law):
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Two forces acting on a particle may be replaced
with a single resultant force, obtained by drawing
the diagonal of a parallelogram which has sides
equal to the given forces.
Figure 1: two forces P and Q
applied to particle A
Figure 2: the resultant force
R obtained graphically from
the parallelogram rule
Vectors add “Head to tail”
Forces at knee joint during traction
Vector Subtraction with the
Parallelogram Law
Figure 1: two forces P and Q
Law of Sines and Law of Cosines
Law of Cosines
a 2 = b 2 + c 2 ! 2bc " cosA
b 2 = a 2 + c 2 ! 2ac " cosB
c 2 = a 2 + b 2 ! 2ab " cosC
Law of Sines
a
b
c
=
=
sinA sinB sinC
Where A, B, and C are the interior angles
of a general triangle, and a, b, and c are
lengths of the sides opposite those angles.
Figure 2: the difference P-Q
obtained graphically as P + (-Q)
from the parallelogram law
Coordinate systems
Different coordinate
systems exist to define the
position of a point P in
space:
Cartesian: x, y, and z
Cylindrical: r, !, and z
Spherical: ", !, and #
Coordinate Systems
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x = r cos !
y = r sin !
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r = x2 + y2
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# y&
! = tan "1 % (
$ x'
Centre of pressure
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Equations of static
equilibrium are
used to determine
location on surface
of force plate where
equivalent resultant
force is applied
(more on this two
lectures from now)
transducers that
measure x, y, and z
components of applied
force and moment
often embedded flush
with the floor
sample surface
dimensions: 40 x 60
cm, 60 x 90 cm
Vertical ground reaction forces
during walking and running
Vertical Force (BW)
Converting between
cartesian and polar:
Force Platforms
3
Walk
Run
2
1
0 0.0
0.1
0.2
0.3
0.4
Time (seconds)
0.5
0.6
0.7
Frictional Forces
How much force must be applied to move
an object?
Frictional forces are
tangential forces that
resist sliding between
contacting surfaces.
W
Fk = µ k N
P
Fm
frictional force F
Frictional forces can exist
with or without
movement (indeed they
often act to prevent
movement!)
Fm=µsN
N
Fk
Static
equilibrium
Motion
applied tangential force P
Review Questions
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what are the four base units in mechanics?
what are the units of force?
what is the difference between a vector and a
scalar?
how do you resolve vectors into their x- and
y-components?
how do you add vectors?
what factors determine the ability of friction
to resist movement?
F
Frictional force F
equals applied
tangential force
P...up to a limit Fm
Fm depends on:
• coefficient of static
friction µs
• normal force N at
interface