Announcements

1
Announcements

• Homework 10 due at end of class Wednesday,
  November 30
• We will meet this Wednesday (November 23)
• Skylab videos showing conservation of angular
  momentum while floating in space
• Symmetry principles in physics

2
Equilibrium

• Much of our investigations so far have been to
  understand the dynamics or motion of an
  object or systems of objects
• Many objects are designed to be static, always
  staying in one place
• Example bridges and buildings
• Such objects are said to be in static
  equilibrium
• Condition for equilibrium is an object or system
  of objects is that they not be accelerated
• Static equilibrium imposes the further
  restriction that the object is at rest

3
Stable and Unstable Equilibrium

• Two cases for static equilibrium
• Unstable Equilibrium the forces and torques sum
  to zero, but a small perturbation would cause the
  system to go out of equilibrium
• Example balancing a pencil on its point
• In principle, it is possible just make all the
  forces and torques sum to zero
• In practice, it is impossible the slightest
  perturbation causes a force or torque that
  displaces the object further and further from
  equilibrium
• Example ball resting on top of a frictionless
  sphere
• Stable Equilibrium the forces and torques sum to
  zero, and a small perturbation would not cause
  the system to go out of equilibrium
• Example pencil lying horizontally on two fingers
• Example ball resting in the bottom of a
  hemispherical shell

4
Stable and Unstable Equilibrium

• Stability is often a matter of degree
• An object that is stable under small
  perturbations may become unstable under large
  perturbations
• Example balancing a quarter on its side
• Its possible to do if one has the patience and
  steady hands
• Small air currents wont knock it over, but
  blowing on it will
• Why is a quarter on its side nearly unstable,
  while one laying flat is very stable?
• Example tipping blocks
• Small amount of tip is stable
• If the tip becomes too large, then unstable
• Calculate torque about support point (x0)
• Does torque act to stabilize block?
• Stable for xcog lt 0, unstable for xcog ? 0

center of gravity
y
center of gravity
x
5
Center of Gravity

• In tipping block example, we treated
  gravitational force as acting at the center of
  gravity position
• As you might guess, this is the same as the
  center of mass
• While we have shown COM motion/dynamics are the
  same as for a point, we need to show that this
  applies to torques as well
• Calculate net torque for object made up of many
  pieces

mi
6
Stable Equilibrium for a Point Support

• If there is a single point of support,
  equilibrium COM will lie along a vertical line
  containing the point of support
• No torques for this point since gravitational
  force is parallel to COM position vector
• Equilibrium will be stable if displacements would
  move the COM position upwards
• Moving COM position upwards would require doing
  work on the object
• Gravitational force will always try to move COM
  to its lowest position
• Example Balanced bird
• Example Curious George
• Example Ramp and Wheel

7
General Approach to Statics Problems

• All components of net forces must sum to 0
• All components of net torque must sum to 0
• Use these equations to solve for unknown/normal
  forces
• Typically, problems will have forces, COM in x-y
  plane
• No z-component of net force
• No x, y components of net torque (zCOM 0,
  Fnet-z 0)

8
Example Walk the Plank

• What are the support forces?
• At what point does plank tip?
• Take origin at right support
• Tips when left support force goes to 0

FR
FL
Mg
mg
D
9
Example Wine Butler

• Torques must balance
• Take origin to be support point