Transcript Lecture 25.Equilibri..

Equilibrium
Lecturer:
Professor Stephen T. Thornton
Reading Quiz
A box is placed on a ramp in the
configurations shown below. Friction
prevents it from sliding. The center of
mass of the box is indicated by a blue dot
in each case. In which case(s) does the
box tip over?
1
A) all
B) 1 only
C) 2 only
D) 3 only
E) 2 and 3
2
3
Reading Quiz
A box is placed on a ramp in the
configurations shown below. Friction
prevents it from sliding. The center of
mass of the box is indicated by a blue dot
in each case. In which case(s) does the
box tip over?
A) all
B) 1 only
C) 2 only
D) 3 only
E) 2 and 3
The torque due to gravity acts
like all the mass of an object is
concentrated at the CM.
Consider the bottom right corner
of the box to be a pivot point.
If the box can rotate such that
the CM is lowered, it will!!
1
2
3
Last Time
Angular momentum and torque
Gyroscopes
Inertial forces, pseudoforces
Review
Today
Force equilibrium
Rotational equilibrium
Stability and balance
Elasticity; stress and strain
The Conditions for (force) Equilibrium
F  0
i
i
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The Conditions for (torque) Equilibrium
The second condition of equilibrium is that
there be no torque around any axis; the choice
of axis is arbitrary.


0
i
i
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Forces Required for Equilibrium
F1  F2  mg force equilibrium
Is the plank in equilibrium?
The forces may be balanced, but is
the torque balanced? Maybe not.
 F  0,  F
Condition for rotational equilibrium:   0
Condition for force equilibrium:
x
y
0
i
Look carefully at
plank problem.
Make sure you
understand how to
get the torques.
Find torque about
any pivot.
3L
t = F2 L - mg
= 0
4
A Lamp in Equilibrium
f y  mg ;
f x  T forces
TV  mgH torque
Do rotational equilibrium demo.
Zero Torque and Balance
This is primarily a torque
problem. Why is that?
Zero Torque and Balance
T = m1g + m2 g
This is primarily a torque
problem. Why is that?
m1gx1  m2 gx2
The Geometric Center of the United States
Do CM of Virginia demo again.
Conceptual Quiz:
A box, with its CM off center as indicated
by the dot, is placed on an inclined plane.
In which of the four orientations shown, if
any, does the box tip over?
A)
B)
C)
D)
E)
A
B
C
D
Box does not tip
over.
·
·
A
·
C
B
·
D
Answer: C
This is the only
orientation where
the CM is to the
left of the likely
pivot point on the
bottom left corner
of the box.
·
·
·
·
Conceptual Quiz
A)
1
4
kg
1-m long. If the system balances at a
B)
1
2
kg
point on the rod 0.25 m from the end
C) 1 kg
holding the mass, what is the mass of
D) 2 kg
the rod?
E) 4 kg
A 1-kg ball is hung at the end of a rod
1m
1kg
Conceptual Quiz
A)
1
4
kg
1-m long. If the system balances at a
B)
1
2
kg
point on the rod 0.25 m from the end
C) 1 kg
holding the mass, what is the mass of
D) 2 kg
the rod?
E) 4 kg
A 1-kg ball is hung at the end of a rod
The total torque about the pivot
must be zero !!
The CM of the
same distance
rod is at its center, 0.25 m to the
X
right of the pivot. Because this
must balance the ball, which is
the same distance to the left of
the pivot, the masses must be
the same !!
mROD = 1 kg
1 kg
CM of rod

1 kg
0.5 kg
Answer: C
0.25 m of stick on each side cancels
out. Next 0.5 m of stick must balance
rock. Distance is 2X, mass in second
half of stick must be ½ kg. Total mass
of stick is then 1 kg.
Solving Statics Problems
1. Choose one object at a time, and make a freebody diagram by showing all the forces on it
and where they act.
2. Choose a coordinate system and resolve
forces into components.
3. Write equilibrium equations for the forces.
4. Choose any axis perpendicular to the plane of
the forces and write the torque equilibrium
equation. A clever choice here can simplify the
problem enormously.
5. Solve.
Solving Statics Problems
If a force in your solution comes out negative (as
FA will here), it just means that it’s in the opposite
direction from the one you chose. This is trivial
to fix, so don’t worry about getting all the signs
of the forces right before you start solving.
Propped Board. A large 62.0-kg board is
propped at a 45° angle against the edge of a
barn door that is 2.6 m wide. How great a
horizontal force must a person behind the
door exert (at the edge) in order to open it?
Assume that there is negligible friction
between the door and the board but that the
board is firmly set against the ground.
Stability and Balance
If the forces on an object are such that they
tend to return it to its equilibrium position, it is
said to be in stable equilibrium.
Stability and Balance
If, however, the forces tend to move it away from
its equilibrium point, it is said to be in unstable
equilibrium.
Stability and Balance
An object in stable equilibrium may become
unstable if it is tipped so that its center of
gravity is outside the pivot point. Of course, it
will be stable again once it lands!
Copyright © 2009 Pearson Education, Inc.
Stability and Balance
People carrying heavy loads automatically
adjust their posture so their center of mass is
over their feet. This can lead to injury if the
contortion is too great.
Copyright © 2009 Pearson Education, Inc.
Demos: CM toys
Do demo of blocks on edge of table.
Conceptual Quiz
A) 5 kg
A (static) mobile hangs as shown
below. The rods are massless and
B) 6 kg
have lengths as indicated. The mass
C) 7 kg
of the ball at the bottom right is 1 kg.
D) 8 kg
What is the total mass of the mobile?
E) 9 kg
?
1m
2m
?
1 kg
1m
3m
Conceptual Quiz
A) 5 kg
A (static) mobile hangs as shown
below. The rods are massless and
B) 6 kg
have lengths as indicated. The mass
C) 7 kg
of the ball at the bottom right is 1 kg.
D) 8 kg
What is the total mass of the mobile?
E) 9 kg
Use torques in two steps: (1)
find the big mass on the bottom
?
left (lower rod only), and (2) use
the entire lower rod assembly
(with two masses) to find the
mass on top right. Finally, add
up all the masses.
1m
2m
?
1 kg
1m
3m
Conceptual Quiz
Consider the two configurations of
A) case 1 will tip
books shown below. Which of the
B) case 2 will tip
following is true?
C) both will tip
D) neither will tip
1
2
1/4
1/2
1/2
1/4
Conceptual Quiz
Consider the two configurations of
A) case 1 will tip
books shown below. Which of the
B) case 2 will tip
following is true?
C) both will tip
D) neither will tip
The CM of the system is
midway between the CM of
1
2
each book. Therefore, the
CM of case #1 is not over the
table, so it will tip.
1/4
1/2
1/2
1/4