Goals:

1
Lecture 21

• Goals

• Discussion conditions for static equilibrium
• Use Free Body Diagrams prior to problem solving
• Introduce Youngs, Shear and Bulk modulus

Exam 3 Wednesday, April, 18th 715-845 PM
2
Statics
Equilibrium is established when
In 3D this implies SIX expressions (x, y z)
3
Three force and three torque expressions
4
Free Body Diagram Rules

• For forces zero acceleration reflects the motion
  of the center of mass
• For torques the position of the force vector
  matters
• We are free to choose the axis of rotation
  (usually one that simplifies the algebra)

5
Statics Using Torque

• Now consider a plank of mass M suspended by two
  strings as shown.
• We want to find the tension in each string

6
Statics Using Torque

• Now consider a plank of mass M suspended by two
  strings as shown.
• We want to find the tension in each string

7
Statics Using Torque

• Now consider a plank of mass M suspended by two
  strings as shown.
• Using a different torque position

8
Another Example See-saw
60 kg
30 kg
30 kg

• Two children (60 kg and 30 kg) sit on a 3.0 m
  long uniform horizontal teeter-totter of mass 30
  kg. The larger child is 1.0 m from the pivot
  point while the smaller child is trying to figure
  out where to sit so that the teeter-totter
  remains motionless.
• Position of the small boy?

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Example Soln.
30 kg
d

• Draw a Free Body diagram (assume g 10 m/s2)
• 0 300 d 300 x 0.5 N x 0 600 x 1.0
• 0 2d 1 4
• d 1.5 m from pivot point

10
Exercise Statics

• A 1.0 kg ball is hung at the end of a uniform rod
  exactly 1.0 m long. The system balances at a
  point on the rod 0.25m from the end holding the
  mass.
• What is the mass of the rod?

(a) 0.5 kg (b) 1 kg (c) 2 kg

1m
1kg
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The classic ladder problem

• A uniform ladder of length L and mass m is
  propped up against a frictionless wall. There is
  friction with the ground with static coefficient
  m. The angle the ladder makes with the ground
  is q.
• What is the minimum value of m to keep the latter
  from slipping?

12
The classic ladder problem

• A uniform ladder of length L and mass m is
  propped up against a frictionless wall. There is
  friction with the ground with static coefficient
  m. If the angle the ladder makes with the
  ground is q.
• What is the minimum value of m that keeps the
  ladder from slipping?

13
Example Statics

• 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
  white dot in each case.
• In which case(s) does the box tip over ?

(a) all (b) 2 3 (c) 3 only

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A stable equilibrium

• Potential energy with tipping (left and right)
• x-axis corresponds to the x-position of the
  center-of mass

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Balancing Acts

• So, where is the center of mass for this
  construction?
• Between the arrows!

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Example problem

• The board in the picture has an average length of
  1.4 m, mass 2.0 kg and width 0.14 m. The angle
  of the board is 45. The bottle is 1.5 m long, a
  center of mass 2/5 of the way from the bottom and
  is mounted 1/5 of way from the end. If the
  sketch represents the physics of the problem.
• What are the minimum and maximum masses of the
  bottle that will allow this system to remain in
  balance?

-0.1 m
-0.8 m
0.0 m
-0.5 m
0.1 m
0.4 m
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Example problem

• What are the minimum and maximum masses of the
  bottle that will allow this system to remain in
  balance?
• X center of mass,
• x, -0.1 m lt x lt 0.1 m

2.0 kg
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Example problem

• What are the minimum and maximum masses of the
  bottle that will allow this system to remain in
  balance?
• Check torque condition (g 10 m/s2)
• x, -0.1 m lt x lt 0.1 m

2.0 kg
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Real Physical Objects

• Representations of matter
• Particles No size, no shape ? can only
  translate along a path
• Extended objects are collections of point-like
  particles
• They have size and shape so, to better reflect
  their motion, we introduce the center of mass
• Rigid objects Translation Rotation
• Deformable objects
• Regular solids
• Shape/size changes under stress (applied
  forces)
• Reversible and irreversible deformation
• Liquids
• They do not have fixed shape
• Size (aka volume) will change under stress

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For Thursday

• All of chapter 12

21
Some definitions

• Elastic properties of solids

• Youngs modulus measures the resistance of a
  solid to a change in its length.
• Bulk modulus measures the resistance of
  solids or liquids to changes in their volume.

elasticity in length
volume elasticity