1. Chapter 12
Equilibrium 1

2. We often want objects to be stable despite forces acting on them
Equilibrium
We often want objects to be stable despite forces acting on them
Consider a book resting on a table, a puck sliding with constant velocity, a rotating ceiling fan, a rolling bicycle wheel with constant velocity
These objects have the characteristics that:
The linear momentum of the center of mass is constant
The angular momentum about the center of mass, or any other point, is constant 2

3. Such objects are in equilibrium
In this chapter we are largely concerned with objects that are not moving at all; P = L = 0
These objects are in static equilibrium
The only one of the examples from the previous page in static equilibrium is the book at rest on the table
Eq. (12-1) 3

4. If a small displacement ends equilibrium, it is unstable
As discussed in 8-3, if a body returns to static equilibrium after a slight displacement, it is in stable static equilibrium
If a small displacement ends equilibrium, it is unstable
Despite appearances, this rock is in stable static equilibrium, otherwise it would topple at the slightest gust of wind
Figure 12-1 4

5. In part (a) of the figure, we have unstable equilibrium
A small force to the right results in (b)
In (c) equilibrium is stable, but push the domino so it passes the position shown in (a) and it falls
The block in (d) is even more stable
Figure 12-2 5

6. Therefore there are 2 conditions for equilibrium:
Requirements for equilibrium are given by Newton's second law, in linear and rotational form
Therefore there are 2 conditions for equilibrium:
Eq. (12-3)
Eq. (12-5) 6

7. 4. The angular momentum of the body L must be zero.
Equilibrium
We often simplify matters by considering forces only in the xy plane, giving:
Note that for static equilibrium we have the additional requirements that:
4. The angular momentum of the body L must be zero.
Eq. (12-7)
Eq. (12-8)
Eq. (12-9) 7

8. Equilibrium
Answer: (c), (e), (f) 8

9. We can simplify this by saying:
Equilibrium
The gravitational force on a body is the sum of gravitational forces acting on individual elements (atoms) of the body
We can simplify this by saying:
Until now we have assumed that the gravitational force acts at the center of mass
This is approximately true for the everyday case: 9

10. Substitute migi for Fgi
Equilibrium
We can show this by considering a sum of torques on each element vs. the torque caused by the gravitational force at the cog
Substitute migi for Fgi
Cancel g (= gi for all i)and divide by the total mass, leaving:
The term on the right is the com
Eq. (12-16)
Figure 12-4 10

11. Some Examples of Static Equilibrium
This section consists of example problems, for forces in the xy plane
Answer: (a) No (b) place the rotation axis at the location where F1 is applied to the beam (c) 45 N 11

12. Static Equilibrium
Previously, when we considered situations involving equilibrium, we assumed that any force on an object acted directly through the center of mass. We can see that if we consider an extended object, forces not at the center of mass can produce not only translational motion, but also rotational motion.
In addition to the condition that ΣF = 0, we need a 2nd condition for equilibrium: Στ = 0. The first conditions is necessary for translational equilibrium, the second is necessary for rotational equilibrium.
Torque can be clockwise or counterclockwise about an arbitrary point of rotation. In order for an object not to accelerate, the torques in either direction around that point must be balanced.
We will consider clockwise torque to be positive.

13. Static Equilibrium
On a simple teeter totter, two individuals can easily produce the conditions for translational and rotational equilibrium. When this happens, the net force is zero and the torque in the clockwise direction will equal the torque in the counterclockwise direction. Equations can be created from the diagram that could then be solved for an unknown quantity.
When writing the equation for rotational equilibrium, an arbitrary location must be chosen for the axis of rotation. Rotation need not occur at that point. Choose the point at a location of an unknown force. This will effectively eliminate it from the equation, but any force exerted at the axis of rotation will not contribute any torque because the lever arm will be zero.

14. Sample Problem 12.01
M = 2.7 kg, m = 1.8 kg 14

15. Sample Problem 12.02
M = 430 kg, m = 85 kg, a = 1.9 m, b = 2.5 m 15

16. Sample Problem 12.03

17. Sample Problem 12.04
R = 9.8 m, h = 60 m, θ = 5.5° 17

18. 12-3 Elasticity
For problems in the xy plane we have 3 independent equations
Therefore we can solve for 3 unknowns
If we have more unknown forces, we cannot solve for them and the situation is indeterminate
This assumes that bodies are rigid and do not deform (there are no such bodies)
With some knowledge of elasticity, we can solve more problems 18

19. 12-3 Elasticity
Answer: (d) 19

20. 12 Summary Static Equilibrium Center of Gravity Elastic Moduli
If the gravitational acceleration is the same for all elements of the body, the cog is at the com.
Eq. (12-3)
Eq. (12-5)
Elastic Moduli
Three elastic moduli
Strain: fractional length change
Stress: force per unit area
Tension and Compression
E is Young's modulus
Eq. (12-23)
Eq. (12-22) 28

21. Lab - Apparatus Bin should contain: knife-edge clamp
4 loops (keep track of these, they are easy to lose)
100-g mass (or 2 50-g masses)
Unknown mass (film canister)
Thumb screw on clamp points down (do not over tighten)
Place base on 2 text books to avoid masses touching table

22. Lab – Procedure
Read and Follow All Directions Do not let masses drop All positions should be stated to nearest 0.001m Follow Part 5 procedure carefully and be sure to record all necessary information, including # & mass of Unknown Work quickly and efficiently Record measurements only today – perform calculations later If you don’t finish today, you must come in outside of class Use the same apparatus Stand so you both see the same side

23. Lab – Using Balance Do not move the balance
Always zero the balance before using
Return balance to equilibrium when finished
Measure to the maximum precision allowed by the balance
kg (5 decimal places)
Mass of meter stick can be found at any time
Be sure to find mass of Unknown for Part 5
Record # and mass of Unknown

24. Lab – Report Information
Use Pencil and write neatly (an unreadable answer is a wrong answer)
Each is responsible for his/her own report
Use g = 9.80 m/s2 as per instructions
Show 1 sample of every unique calculation
Use GUESS method
Counterclockwise torque is (+), clockwise torque is (-)
Answer questions in complete sentences
Support your answers with appropriate evidence