Chapter 12 – Static equilibrium
and Elasticity
Lecture 1
April 13, 2009
•  Conditions for static equilibrium
•  Center of gravity
•  Equilibrium in an accelerated frame
4/13/10
Physics 201, Spring 2010, U. Wisconsin
1
Equilibrium
•  To ensure mechanical equilibrium, you need to ensure
translational equilibrium as well as rotational
•  The First Condition of Equilibrium states
–  The net external force must be zero
•  The Second Condition of Equilibrium states
–  The net external torque must be zero
4/13/10
Physics 201, Spring 2010, U. Wisconsin
2
Notes About Equilibrium

A zero net force does not mean the absence of translational motion:
F = 0  a = 0.
v = 0  static equilibrium;
v = const  dynamic equilibrium

A zero net torque does not mean the absence of rotational motion:
  An object that rotates at uniform angular velocity can be under the influence
of a zero net torque (dynamic equilibrium)
4/13/10
Physics 201, Spring 2010, U. Wisconsin
3
A car is parked on a steep hill facing downwards with tires turned
towards the curb as they should be. Is the car in dynamic
equilibrium.
a) yes
b) no
Equilibrium means the net force and
net torque vanish (= zero).
This means constant velocity
V = 0 (STATIC)
V = constant vector (DYNAMIC)
The car is in static equilibrium.
4/13/10
Physics 201, Spring 2010, U. Wisconsin
4
A car is running on cruise control on a country road along rolling
hills. Assuming that the cruise control does a perfect job, is the
car in dynamic equilibrium.
a) yes
b) no
Force is needed to move over the hills even
at constant speed.
Also needed to move on flat curves.
4/13/10
Physics 201, Spring 2010, U. Wisconsin
5
Solving Equilibrium Problems
•
Draw a diagram of the system
Isolate the object being analyzed and draw a free body diagram showing all the
external forces acting on the object
–  For systems containing more than one object, draw a separate free body
diagram for each object
•
Establish convenient coordinate axes for each object.
–  Apply the First Condition of Equilibrium Fnet = 0.
•
Choose a convenient rotational axis for calculating the net torque on the object.
–  Apply the Second Condition of Equilibrium τnet = 0.
(not all equations are independent; choose simple one cleverly.)
•
Solve the resulting simultaneous equations for all of the unknowns
4/13/10
Physics 201, Spring 2010, U. Wisconsin
6
Axis of Rotation
•  If the object is in equilibrium, it does not matter where you
put the axis of rotation for calculating the net torque
–  When solving a problem, you must specify an axis of
rotation, and maintain that choice consistently
throughout the problem
–  The location of the axis of rotation is completely
arbitrary
–  Often the nature of the problem will suggest a
convenient location for the axis: Would be good to
choose the axis with more unknown forces going
through it, so that those torques all vanish.
4/13/10
Physics 201, Spring 2010, U. Wisconsin
7
A SPECIAL POINT
We introduced the center of mass
In uniform gravitational field (near earth surface),
the center of mass   center of gravity:
4/13/10
Physics 201, Spring 2010, U. Wisconsin
8
Experimentally Determining the
Center of Gravity
•  The wrench is hung freely
from two different pivots
•  The intersection of the
lines indicates the center
of gravity
•  A rigid object can be
balanced by a single force
equal in magnitude to its
weight as long as the force
is acting upward through
the object’s center of
gravity
4/13/10
Physics 201, Spring 2010, U. Wisconsin
9
Can the center of gravity be outside of the object?
Where is the center of gravity of a “yummy” donut?
It is at the origin of the circular ring, half way from the
bottom of the donut - where there is no dough.
4/13/10
Physics 201, Spring 2010, U. Wisconsin
10
A moment later……
C G has shifted along the line of symmetry away from the bite.
4/13/10
Physics 201, Spring 2010, U. Wisconsin
High Jump
Olympic high jumpers
(including all high
jumpers) manipulate
things so that their
body goes over the
bar -- but their c.g.
goes under !!
(so to lower potential
energy, thus initial
kinetic energy.)
4/13/10
Physics 201, Spring 2010, U. Wisconsin
More High Jumping
We know how to find the
motion of an object with
initial velocity
After leaving the ground
only gravity acts
In order to win, the jumper
must wiggle his/her
body around the cg
(rotation)
4/13/10
Physics 201, Spring 2010, U. Wisconsin
13
Question
Pick free body diagram corresponding to the picture
below assuming that the person is able to hold the pole
in equilibrium:
FL
A
F
FR FL
R
FL
B
4/13/10
W
C
F
R
W
Physics 201, Spring 2010, U. Wisconsin
W
14
Example of a
Free Body Diagram
•
•
4/12/10
The free body diagram includes the
directions of the forces
The weights act through the centers
of gravity of their objects
Physics 201, Spring 2010, U. Wisconsin
15
4/12/10
Physics 201, Spring 2010, U. Wisconsin
16
4/12/10
Physics 201, Spring 2010, U. Wisconsin
17
4/12/10
Physics 201, Spring 2010, U. Wisconsin
18
4/12/10
Physics 201, Spring 2010, U. Wisconsin
19
The object shown in the diagram is a cube of uniform density
resting on a rough surface. The applied force F is balanced by
the frictional force F fr. When the block is on the verge of
tipping, the point of application of the normal force acting on
the cube will be
A.
B.
C.
D.
E.
1
2
3
4
5
The object shown in the diagram is a cube of uniform density
resting on a rough surface. The applied force F is balanced by
the frictional force F fr. When the block is on the verge of
tipping, the point of application of the normal force acting on
the cube will be
A.
B.
C.
D.
E.
1
2
3
4
5
Condition for static equilibrium
in accelerated frame
•  Conditions for a rigid body to be in a static
equilibrium in a linearly accelerated frame, eg a
truck moving a block.
A)  Net external force must be modified to account for
the linear acceleration, by
a “fictitious force”
B)  Net external torque must be
still 0: no angular acceleration
4/12/10
Physics 201, Spring 2010, U. Wisconsin
22