Chapter 12 – Static equilibrium and Elasticity

Lecture 1

•  Conditions for static equilibrium •  Center of gravity •  Equilibrium in an accelerated frame

April 13, 2009

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Equilibrium

•  To ensure mechanical equilibrium, you need to ensure translational equilibrium as well as rotational

•  The First Condition of Equilibrium states –  The net external force must be zero

•  The Second Condition of Equilibrium states –  The net external torque must be zero

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Notes About Equilibrium   A zero net force does not mean the absence of translational motion: F = 0 a = 0. v = 0 static equilibrium; v = const dynamic equilibrium

  A zero net torque does not mean the absence of rotational motion:   An object that rotates at uniform angular velocity can be under the influence

of a zero net torque (dynamic equilibrium)

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A car is parked on a steep hill facing downwards with tires turned towards the curb as they should be. Is the car in dynamic equilibrium.

a) yes b) no

The car is in static equilibrium.

Equilibrium means the net force and net torque vanish (= zero).

This means constant velocity V = 0 (STATIC) V = constant vector (DYNAMIC)

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A car is running on cruise control on a country road along rolling hills. Assuming that the cruise control does a perfect job, is the car in dynamic equilibrium. a) yes b) no

Force is needed to move over the hills even at constant speed.

Also needed to move on flat curves.

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Solving Equilibrium Problems •  Draw a diagram of the system Isolate the object being analyzed and draw a free body diagram showing all the

external forces acting on the object –  For systems containing more than one object, draw a separate free body

diagram for each object

•  Establish convenient coordinate axes for each object. –  Apply the First Condition of Equilibrium Fnet = 0.

•  Choose a convenient rotational axis for calculating the net torque on the object. –  Apply the Second Condition of Equilibrium τnet = 0.

(not all equations are independent; choose simple one cleverly.)

•  Solve the resulting simultaneous equations for all of the unknowns

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Axis of Rotation

•  If the object is in equilibrium, it does not matter where you put the axis of rotation for calculating the net torque –  When solving a problem, you must specify an axis of

rotation, and maintain that choice consistently throughout the problem

–  The location of the axis of rotation is completely arbitrary

–  Often the nature of the problem will suggest a convenient location for the axis: Would be good to

choose the axis with more unknown forces going through it, so that those torques all vanish.

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A SPECIAL POINT

We introduced the center of mass

In uniform gravitational field (near earth surface), the center of mass center of gravity:

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Experimentally Determining the Center of Gravity

•  The wrench is hung freely from two different pivots

•  The intersection of the lines indicates the center of gravity

•  A rigid object can be balanced by a single force equal in magnitude to its weight as long as the force is acting upward through the object’s center of gravity

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Can the center of gravity be outside of the object?

Where is the center of gravity of a “yummy” donut?

It is at the origin of the circular ring, half way from the bottom of the donut - where there is no dough.

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A moment later……

C G has shifted along the line of symmetry away from the bite.

High Jump

Olympic high jumpers (including all high jumpers) manipulate things so that their body goes over the bar -- but their c.g. goes under !!

(so to lower potential energy, thus initial kinetic energy.)

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More High Jumping

We know how to find the motion of an object with initial velocity

After leaving the ground only gravity acts

In order to win, the jumper must wiggle his/her body around the cg (rotation)

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Question Pick free body diagram corresponding to the picture

below assuming that the person is able to hold the pole in equilibrium:

W F R

F L

B

W

F R F L

C

W F R

F L

A

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Example of a Free Body Diagram

•  The free body diagram includes the directions of the forces

•  The weights act through the centers of gravity of their objects

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The object shown in the diagram is a cube of uniform density resting on a rough surface. The applied force F is balanced by

the frictional force F fr. When the block is on the verge of tipping, the point of application of the normal force acting on

the cube will be

A.  1 B.  2 C.  3 D.  4 E.  5

The object shown in the diagram is a cube of uniform density resting on a rough surface. The applied force F is balanced by

the frictional force F fr. When the block is on the verge of tipping, the point of application of the normal force acting on

the cube will be

A.  1 B.  2 C.  3 D.  4 E.  5

Condition for static equilibrium in accelerated frame

•  Conditions for a rigid body to be in a static equilibrium in a linearly accelerated frame, eg a truck moving a block.

A)  Net external force must be modified to account for the linear acceleration, by a “fictitious force”

B)  Net external torque must be still 0: no angular acceleration

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