Torque and static equilibrium

1
Chapter 8 Equilibrium and Elasticity
Topics

• Torque and static equilibrium
• The spring force
• Hookes law
• Elastic materials
• The elastic limit

Sample question
How does a dancer balance so gracefully en
pointe? And how does her foot withstand the great
stresses concentrated on her toes?
Slide 8-1
2
Torque and Static Equilibrium
For an extended object to be in equilibrium, the
net force and the net torque must be zero.
Slide 8-8
3
Choosing the Pivot Point
Slide 8-9
4
Solving Static Equilibrium Problems
Slide 8-10
5
Checking Understanding

• What does the scale read?
• 500 N
• 1000 N
• 2000 N
• 4000 N

Slide 8-11
6
Answer

• What does the scale read?
• 2000 N

Slide 8-12
7
Example
A 2-m-long board weighing 50 N extends out over
the edge of a table, with 40 of the boards
length off the table. How far beyond the table
edge can a 25 N cat walk before the board begins
to tilt?
0.8 m
Slide 8-13
8
Example
A 2-m-long board weighing 50 N extends out over
the edge of a table, with 40 of the boards
length off the table. How far beyond the table
edge can a 25 N cat walk before the board begins
to tilt?
pivot
0.8 m
Slide 8-13
9
Example
A 2-m-long board weighing 50 N extends out over
the edge of a table, with 40 of the boards
length off the table. How far beyond the table
edge can a 25 N cat walk before the board begins
to tilt?
n
Wcat 25 N
Wboard 50 N
What is the magnitude of n?
10
Example
A 2-m-long board weighing 50 N extends out over
the edge of a table, with 40 of the boards
length off the table. How far beyond the table
edge can a 25 N cat walk before the board begins
to tilt?
0.8 m
n
X
0.2 m
Wcat 25 N
Wboard 50 N
What is the magnitude of n? n 75 N.
11
What is n when the cat is right over the center
of the board?
0.8 m
0.2 m
Wcat 25 N
Wboard 50 N
12
What is n when the cat is right over the center
of the board? n 50 N (board) 25 N (cat) 75
N.
0.8 m
0.2 m
Wcat 25 N
Wboard 50 N
13
What is n when the cat is between the boards
center and the table edge?
0.8 m
0.2 m
Wcat 25 N
Wboard 50 N
14
What is n when the cat is between the boards
center and the table edge? The (board cat) CM
moves to a point between the center and the table
edge, and the joint 75 N weight is balanced by a
75 N normal force.
0.8 m
n 75 N
0.2 m
Wcat 25 N
Wboard 50 N
15
As the cat walks past the table edge, there is a
critical distance (x) when the board just begins
to tip. At this point the normal force (n) is
still 75 Nit has moved to the pivot point at the
edge of the table.
0.8 m
n 75 N
X
0.2 m
Wcat 25 N
Wboard 50 N
16
Lets find the non-zero torques about ( ) just
at the point of the board about to start tipping
0.8 m
n 75 N
X
0.2 m
Wcat 25 N
Wboard 50 N
17
Lets find the non-zero torques about ( ) just
at the point of the board about to start tipping
define CCW as -
0.8 m
n 75 N
X
0.2 m
Wcat 25 N
Wboard 50 N
18
Lets find the non-zero torques about ( ) just
at the point of the board about to start tipping
define CCW as -
0.8 m
n 75 N
X
0.2 m
Wcat 25 N
Wboard 50 N
Equilibrium condition
19
So, (-10 Nm) (25 N) x 0 N, or x (10
Nm)/(25 N) 0.4 m. The cat is in the middle of
the board section past the edge of the table. A
mm farther and the board suddenly tips!
0.2 m
.41 m
Wcat 25 N
20
Conceptual Question 3 p 256. Could a ladder on
a level floor lean against a wall in static
equilibrium if there were no frictional forces?
Explain.
No friction
21
Question 8-19 p 257. A tall ladder is leaning
against a wall, as shown. There is no friction
between the top of the ladder and the wall. The
coefficient of static friction between the bottom
of the ladder and the ground is small but not
zero. A painter climbs up the ladder to reach a
high spot on the wall. At

• Which location should the painter be most worried
  about the ladder slipping?
• Near the bottom.
• At the middle of the ladder
• Near the top.
• The risk is the same at all locations.

22
Problem 8-4 p 257. How much torque must the
pin exert to keep the rod in the above figure
from rotating? Calculate this torque about a
point where the pin enters the rod and is
perpendicular to the plane of the figure.
23
npin
4.9 N
40 cm
Wrod 19.6 N
24
Problem 36 p 259. An 80 kg construction worker
sits down 2.0 m from the end of a 1450 kg steel
beam to eat his lunch, as shown above. The cable
supporting the beam is rated a 15,000 N. Should
the worker be worried?
25
T
784 N
14,210 N
3 m
1 m
Are there any other forces acting on the beam?
26
Fwall
T
784 N
14,210 N
3 m
1 m
Are there any other forces acting on the beam? Of
course, the wall holds on to the beam.
27
Fwall
T
784 N
14,210 N
3 m
1 m
Are there any other forces acting on the beam? Of
course, the wall holds on to the beam.
28
T
784 N
14,210 N
3 m
1 m
Are there any other forces acting on the beam? Of
course, the wall holds on to the beam.
29
Problem 37 p 259 A forearm can be modeled as a
1.2 kg, 35 cm long beam that pivots at the
elbow and is supported by the biceps, as shown.
How much force must the biceps exert to hold a
500 g ball with the forearm parallel to the floor?
30
Problem 37 p 259 A forearm can be modeled as a
1.2 kg, 35 cm long beam that pivots at the
elbow and is supported by the biceps, as shown.
How much force must the biceps exert to hold a
500 g ball with the forearm parallel to the
floor? Any other forces on the forearm?
Fbiceps
Warm
Wball
31
Problem 37 p 259 A forearm can be modeled as a
1.2 kg, 35 cm long beam that pivots at the
elbow and is supported by the biceps, as shown.
How much force must the biceps exert to hold a
500 g ball with the forearm parallel to the
floor? Any other forces on the forearm? Yes!
Fbiceps
Warm
Wball