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Physics 211
11: Static Equilibrium and
Elasticity
•Conditions for Static Equilibrium for Rigid Objects
•Center of Gravity
•Examples
•Elastic Properties of Objects
Torque associated with a force
Magnitude of the force times
the perpendicular distance
from the line of action of the
force to the pivot point
r
d
F
Conditions for Static
Equilibrium for Rigid Objects
The center of mass of an object does not accelerate
F tot
if the total force on the object is zero i.e.
= 0  acm = 0  TRANSLATIONAL EQUILIBRIUM
An object will have zero angular acceleration if the
total torque on the object is zero i.e
t tot = 0  a = 0  ROTATIONAL EQUILIBRIUM
If the initial velocity of the center of mass is zero
and the initial angular velocity is zero they will
remain zero if
F tot = 0 and a cm = 0
When this is so the object is said to be in
STATIC EQUILIBRIUM
The torque of a force about any point on the
line of action of that force is zero
If a body is in translational equilibrium
the net torque with respect to one point
is the same with respect to ALL points in the body
In particular if it is zero about one point
It is zero about all points!
Total torque about point O is
t O = r 1  F 1 + r 2  F 2 +L + r n  F n =

ri  Fi
i
total torque is taken about another point O 
tO  =
(
is
r i - r)  F i
i
where r  is the displacement vector from O to O 
 t O  =
(
r i  F i - r   F i ) =
i
(
i
Translational equilibrium  F 1 + F 2 +L +F n =
\ t O  =
(
ri  F i ) = tO
i


ri  F i ) - r 
Fi
i
Fi = 0
i
Center of Gravity
1
rcg =
Wtot
W r =
i
i
i
mg r

m g
1
i i
i i
i
i
i
Where the system is made up of discrete objects of
mass mi and the acceleration of gravity at the location
of these masses is gi . If the gravitational field strength
is uniform i.e.does not vary,
the center of gravity is at
the same position as the center of mass
r cg =
1
W tot

i
Wi ri =


1
mi g
mi g r i =
i
i


1
mi
mi ri = r cm
i
i
If system is an extended object the sums are
replaced by integrals.
If an object (extended or many discrete ones rigidly
connected ) are suspended by an upward force in a homogenous
gravitational field then when the object achieves static equilibrium
Ftot = F up + Mtot g = 0 (no change of vcm from 0)  F up = -Mtot g
t tot about center of mass must be zero (no change of acm from 0)
t tot = text = 0 = r 1  F up
where r1 is a displacement vector to the center of gravity

= center of mass
r1 is pointing towards the center of gravity and is parallel to the forc
 Fup acts at a point vertically above the center of mass / gravity
where the total weight vector acts
Elastic Properties of Objects
States of Matter
not
resistant
to forces
gas
liquid
solid
resistant
to
compression
forces
resistant
to
compression
and
shearing
forces
compression
shearing
Elastic Materials = Non rigid
Elasticity is measured
by the response of the
material to an applied
force.
Hookes Law
F = -kd
Restoring force is
proportional to the
displacement from the
equilibrium position
Displacement from the
equilibrium position is
proportional to the applied
force
Stress = [Constant] x Strain
Force F
Stress =
=
= Pressure
Area
A
change in size
Strain =
initial size
change in length
l
for a bar:
=
initial length
l0
for a surface:
change in area
A
=
initial area
A0
change in volume
V
for a volume:
=
initial volume
V0
1 - Dimensional
(Tension / Compression)
F
l
;
= Y
A
l0
3 - Dimensional
P = - B
Y = Youngs Modulus
(Tension / Compression )
V
; B = Bulk Modulus
V0
3 - Dimensional
(Shearing )
F
x
; S = Shear Modulus
= S
A
h