1. Physics 103: Lecture 14 Rotations of Extended Objects
Today’s lecture will cover
Torque
Equilibrium
Moment of Inertia
Torque and Angular Acceleration
03/10/2003
Physics 103, Spring 2004, U. Wisconsin 1

2. Physics 103, Spring 2004, U. Wisconsin
Torque
“Point” objects with mass m are linearly accelerated according to Newton’s 2nd Law.
But what if the objects have size?
Ignore it (treat the mass as a point)
Or, account for it:
object is rigid … all points are fixed but the extended object can rotate
Torque,t, is the tendency of a force to rotate an object about some axis F d a
offset force
03/10/2003
Physics 103, Spring 2004, U. Wisconsin

3. Physics 103, Spring 2004, U. Wisconsin
Torque
03/10/2003
Physics 103, Spring 2004, U. Wisconsin

4. Physics 103, Spring 2004, U. Wisconsin
Lever Arm
The lever arm, d, is the perpendicular distance from the axis of rotation to a line drawn from the direction of the force
d = L sin f
t = FL sin f
03/10/2003
Physics 103, Spring 2004, U. Wisconsin

5. An Alternative Look at Torque
The force could also be resolved into its x- and y-components
The x-component, F cos f, produces 0 torque
The y-component, F sin f, produces a non-zero torque
t = FL sin f L
03/10/2003
Physics 103, Spring 2004, U. Wisconsin

6. Question 1: Opening the door
Torque
Force
Point of application
Distance to center of rotation
+ or - (ccw or cw)
In which case (a) thru (f) is the torque most positive?
03/10/2003
Physics 103, Spring 2004, U. Wisconsin

7. Physics 103, Spring 2004, U. Wisconsin
Equilibrium
To ensure mechanical equilibrium, you need to ensure rotational equilibrium as well as translational
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
03/10/2003
Physics 103, Spring 2004, U. Wisconsin

8. Notes About 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
This is analogous to the translational situation where a zero net force does not mean the object is not in motion
03/10/2003
Physics 103, Spring 2004, U. Wisconsin

9. Physics 103, Spring 2004, U. Wisconsin
Center of Gravity
The force of gravity acting on an object must be considered in determining equilibrium
In finding the torque produced by the force of gravity, all of the weight of the object can be considered to be concentrated at one point
The coordinates of the center of gravity can be found from the sum of the torques acting on the individual particles within the object being set equal to the torque produced by the weight of the object
03/10/2003
Physics 103, Spring 2004, U. Wisconsin

10. 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
03/10/2003
Physics 103, Spring 2004, U. Wisconsin

11. Physics 103, Spring 2004, U. Wisconsin
Preflight 1
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
Equilibrium means the net force and net torque vanish (= zero).
This means constant velocity
V = 0 (STATIC)
V = constant vector (DYNAMIC)
The car is in static equilibrium.
03/10/2003
Physics 103, Spring 2004, U. Wisconsin

12. Physics 103, Spring 2004, U. Wisconsin
Preflight 2
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 velocity.
(Also needed to move on flat curves.)
For example, true in Utah on a straight away on the salt flats.
03/10/2003
Physics 103, Spring 2004, U. Wisconsin

13. Physics 103, Spring 2004, U. Wisconsin
Preflight 3
An old carpenter's trick to keep nails from bending when they are pounded into hard materials is to grip the center of the nail firmly with pliers. Why does this help?
Because, the middle of the nail is its center of gravity. Holding it firmly there improves its static equilibrium.
Any small angle in the hammer striking the nail produces a torque. If the nail is held right in the middle, the torque will be half the value that would occur without the pliers.
Both of the above
None of the above
03/10/2003
Physics 103, Spring 2004, U. Wisconsin

14. Physics 103, Spring 2004, U. Wisconsin
Stability
An object is in equilibrium:
Imagine a small shift (displacement)
Stable -- generated torque restores equilibrium
03/10/2003
Physics 103, Spring 2004, U. Wisconsin

15. Physics 103, Spring 2004, U. Wisconsin
Stability (reprise)
Unstable -- generated torque destroys equilibrium
03/10/2003
Physics 103, Spring 2004, U. Wisconsin

16. 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
Choose a convenient rotational axis for calculating the net torque on the object.
Apply the Second Condition of Equilibrium
Solve the resulting simultaneous equations for all of the unknowns
03/10/2003
Physics 103, Spring 2004, U. Wisconsin

17. Physics 103, Spring 2004, 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
The location of the axis of rotation is completely arbitrary
Often the nature of the problem will suggest a convenient location for the axis
When solving a problem, you must specify an axis of rotation
Once you have chosen an axis, you must maintain that choice consistently throughout the problem
03/10/2003
Physics 103, Spring 2004, U. Wisconsin

18. Physics 103, Spring 2004, U. Wisconsin
Question 2
Considering the figure shown, which of the statements below are true?
Only the First Condition of Equilibrium is satisfied
Only the Second Condition of Equilibrium is satisfied
Both conditions are satisfied
Neither conditions are satisfied
03/10/2003
Physics 103, Spring 2004, U. Wisconsin

19. Physics 103, Spring 2004, U. Wisconsin
Example 1
Free Body Diagram F R F L W
03/10/2003
Physics 103, Spring 2004, U. Wisconsin

20. Physics 103, Spring 2004, U. Wisconsin
Question 3
Pick free body diagram corresponding to the picture below assuming that the person is able to hold the pole in equilibrium: W F R L A W F R L C W F R L B
03/10/2003
Physics 103, Spring 2004, U. Wisconsin

21. Example of Free Body Diagram
Isolate the object to be analyzed
Draw the free body diagram for that object
Include all the external forces acting on the object
03/10/2003
Physics 103, Spring 2004, U. Wisconsin

22. 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
03/10/2003
Physics 103, Spring 2004, U. Wisconsin

23. Physics 103, Spring 2004, U. Wisconsin
High Jump
Olympic high jumpers (including all high jumpers) manipulate things so that their body goes over the bar -- but their c.m. goes under!!
03/10/2003
Physics 103, Spring 2004, U. Wisconsin

24. Physics 103, Spring 2004, 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)
03/10/2003
Physics 103, Spring 2004, U. Wisconsin

25. Physics 103, Spring 2004, U. Wisconsin
Tides
The tides are caused by the action of the moon on the earth.
center of mass of moon-earth system A C B
mac = Fg
At C there is a balance (ac is centripetal acceleration).
At A Gravitation is too weak to provide c.a.
At B Gravity is stronger than necessary to provide c.a.
03/10/2003
Physics 103, Spring 2004, U. Wisconsin