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PreClass Notes: Chapter 12

• From Essential University Physics 3rd Edition

• by Richard Wolfson, Middlebury College

• ©2016 by Pearson Education, Inc.

• Narration and extra little notes by Jason Harlow,

University of Toronto

• This video is meant for University of Toronto

students taking PHY131.

Outline

“The key to the engineer’s success [in

making sure the Alamillo Bridge

doesn’t topple to the left] is static

equilibrium – the condition in which a

structure … experiences neither a net

force nor a net torque.”– R.Wolfson

• 12.1 Conditions for equilibrium

• 12.2 Center of Gravity

• 12.3 Static Equilibrium

Problems

• 12.4 Stability

Image from https://www.bluffton.edu/~sullivanm/spain/seville/calatravabridge/bridge.html ]

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Conditions for Static Equilibrium

• A system in static equilibrium undergoes neither

rotational nor linear acceleration.

– If it’s at rest, it remains at rest.

• The conditions for static equilibrium are:

– No net force:

– No net torque:

• Torques can be evaluated about any

convenient point.

0iF

0i i ir F

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• The figure shows three pairs of forces acting on an object.

Which pair, acting as the only forces on the object, results

in static equilibrium?

Got it?

A. Force pair

B. Force pair

C. Force pair

1 2+A A

1 2+B B

1 2+C C

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© 2012 Pearson Education, Inc. Slide 1-7

Center of Gravity

• The gravitational forces acting on all parts of an

object exert a torque on the object.

– These forces act like a single force, equal to the

object’s weight, acting at a point called the

center of gravity.

– In a uniform gravitational field, the center of

gravity coincides with the center of mass.

© 2012 Pearson Education, Inc. Slide 1-8

Finding the Center of Gravity

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© 2012 Pearson Education, Inc. Slide 1-9

Finding the Center of Gravity

© 2012 Pearson Education, Inc. Slide 1-10

Finding the Center of Gravity

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• The dancer in the figure is balanced; that is, she is in

static equilibrium. Which of the three lettered points

could be her center of gravity?

A. Point A

B. Point B

C. Point C

Got it?

Example 12.2 : At what angle will the ladder slip?

• Frictionless wall.

• Coefficient of

static friction with

the ground is μ.

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Stability

• An equilibrium is stable if a slight disturbance from

equilibrium results in forces and/or torques that tend to

restore the equilibrium.

• An equilibrium is unstable if a slight disturbance causes

the system to move away from the original equilibrium.

Cone on its base is stable Cone on its tip is unstable

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Kinds of Stability

Stable equilibrium: disturbed

ball will return to equilibrium

Metastable or conditionally stable

equilibrium: ball returns for small

disturbances, but not for large ones

Unstable equilibrium:

disturbed ball will leave

original equilibrium

Neutrally stable equilibrium

Conditions for Equilibrium and Stability

• To be in equilibrium, there must be zero net force on an object.

– Therefore the object must be at a maximum or minimum of its

potential energy curve:

• For stable equilibrium, the object must be at a minimum:

• The condition for unstable equilibrium is

dU

dx 0 (condition for equilibrium)

d 2U

dx2 0 (stable equilibrium)

2

20 (unstable equilibrium)

d U

dx

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Conditions for Equilibrium and Stability

In two and three

dimensions, an

object can be

stable in one

direction but not

another.