View
17
Download
0
Category
Preview:
Citation preview
Chapter 12 – Static equilibrium and Elasticity
Lecture 1
• Conditions for static equilibrium • Center of gravity • Equilibrium in an accelerated frame
April 13, 2009
4/13/10 1 Physics 201, Spring 2010, U. Wisconsin
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
4/13/10 2 Physics 201, Spring 2010, U. Wisconsin
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)
4/13/10 3 Physics 201, Spring 2010, U. Wisconsin
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)
4/13/10 4 Physics 201, Spring 2010, U. Wisconsin
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.
4/13/10 5 Physics 201, Spring 2010, U. Wisconsin
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
4/13/10 6 Physics 201, Spring 2010, U. Wisconsin
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.
4/13/10 7 Physics 201, Spring 2010, U. Wisconsin
4/13/10 Physics 201, Spring 2010, U. Wisconsin 8
A SPECIAL POINT
We introduced the center of mass
In uniform gravitational field (near earth surface), the center of mass center of gravity:
4/13/10 Physics 201, Spring 2010, U. Wisconsin 9
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
4/13/10 Physics 201, Spring 2010, U. Wisconsin 10
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.
4/13/10 Physics 201, Spring 2010, U. Wisconsin
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.)
4/13/10 Physics 201, Spring 2010, U. Wisconsin
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)
4/13/10 13 Physics 201, Spring 2010, U. Wisconsin
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
4/13/10 14 Physics 201, Spring 2010, U. Wisconsin
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
4/12/10 15 Physics 201, Spring 2010, U. Wisconsin
4/12/10 16 Physics 201, Spring 2010, U. Wisconsin
4/12/10 17 Physics 201, Spring 2010, U. Wisconsin
4/12/10 18 Physics 201, Spring 2010, U. Wisconsin
4/12/10 19 Physics 201, Spring 2010, U. Wisconsin
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
4/12/10 22 Physics 201, Spring 2010, U. Wisconsin
Recommended