1 7/29/2004 Midterm 2 – Tomorrow (7/30/04)  Material from Chapters 7-12  Room where recitation meets Practice Exam available on-line or in Davey library Some good practice problems  7.13, 8.7, 8.21, 9.10, 9.23, 9.35, 10.4, 10.23, 10.29, 11.33, 11.56, 12.19, 12.23, Announcements

Chapter 13 Equilibrium and Elasticity

3 7/29/2004 Equilibrium An object is in equilibrium if: and Examples: fan blades, ball rolling across the floor… An object is in static equilibrium if it is at rest in our reference frame and Examples: bridges, skyscrapers…

4 7/29/2004 Stable vs. Unstable Static Equilibrium Stable Equilibrium: A small perturbation of the objects position results in the object being pushed back to its original position by restoring forces Unstable Equilibrium: A small perturbation of the objects position results in the object being pushed away from its original position. Neutral Equilibrium: When displaced, the object stays in new position

5 7/29/2004 Types of Equilibrium U x stable unstable neutral

6 7/29/2004 Stable vs Unstable Equilibrium Unstable Equilibrium: easy to knock things over Examples: Dominos or a Rube Goldberg machine

7 7/29/2004 Demo: Equilibrium If I stack two blocks on top of each other, what is the condition for equilibrium? L ½ L Center of mass of top block must be supported by the bottom block

8 7/29/2004 Demo: Equilibrium What about 3 blocks? CM of top two blocks must be supported by bottom block L ½ L ??

9 7/29/2004 Demo: Equilibrium 4 blocks? A pattern is developing…

10 7/29/2004 Demo: Equilibrium This is known as the geometric series, and it never converges! With enough blocks, we can make the overhang as big as we like! With 5 blocks: Complete overhang!

11 7/29/2004 Requirements for Equilibrium If an object is in equilibrium, then: So we can solve statics problems using only the physics we already know!

12 7/29/2004 Problem Solving Strategy: 1)Draw and label a diagram 2)Pick an appropriate origin a)torques sum to zero about any choice of origin 3)Sum forces and torques 4)Solve for unknown quantities

13 7/29/2004 Example: Teeter-Totter mg Mg d=2mD A 80 kg parent and a 20 kg child are balancing on a see-saw. If the child sits 2 m from the pivot, where does the parent need to sit, and what is the force on the pivot?

14 7/29/2004 Example: Pick origin at pivot (makes the torque from the force at the pivot = 0) mgMg d=2mD FpFp Summing the torques: +

15 7/29/2004 Example: mgMg d=2mD FpFp Summing the forces:

16 7/29/2004 Example: Another Way… mgMg d=2mD=0.5m FpFp Pick dad as origin, then sum torques: Same result! +

17 7/29/2004 Example: 10 in 2 in3 in 3 N 2 N 0.8 N 5 in T 1 =?T 2 =? Weights are suspended from a rod which is suspended by two ropes at its ends. Find the tension in these ropes, T 1 and T 2.

18 7/29/2004 Example: (continued) 10 in 8 in 3 in 3 N 2 N 0.8 N 5 in T 1 =?T 2 =? Pick origin at right edge Sum torques: +

19 7/29/2004 Example: (continued) 10 in 8 in 3 in 3 N 2 N 0.8 N 5 in T 1 =?T 2 =? Sum forces:

20 7/29/2004 Does it work? T1T1 T2T2

21 7/29/2004 Standing on a Ladder θ L d F N1 F N2 Mg mg F fr A 5 m ladder rests against a frictionless wall at an angle 30º from vertical. The weight of the ladder is 200 N, uniformly distributed along its length. A 1000 N person climbs the ladder. What coefficient of friction must the ladder have with the floor so the ladder does not slip when the person has climbed 4 m along the ladder?

22 7/29/2004 Standing on a Ladder L d F N1 F N2 Mg mg F fr Pick the origin at the bottom of the ladder: Summing forces in the x-direction: Summing forces in the y-direction:

23 7/29/2004 Standing on a Ladder L d F N1 F N2 Mg mg F fr Summing the torques:

24 7/29/2004 Towing a Trailer Safely d 1 =3m d 2 =5m xd=4m NhNh N1N1 N2N2 ½Mg A trailer hitch has a rated ‘tongue weight’ of 2000 N. A lightweight two-axle trailer, with axles 3 m and 5 m behind the hitch, is pulled behind the vehicle. A 4 m car weighing N is parked on the trailer. How far back must the car be parked, and what are the forces on the axles?

25 7/29/2004 Towing a Trailer Safely d 1 =3m d 2 =5m xd=4m NhNh N1N1 N2N2 ½Mg Sum forces: Pick the hitch as the origin. Sum torques: Don’t know x, N 1, N 2 2 equations and 3 unknowns! Not enough info!

26 7/29/2004 Indeterminate Structures Sometimes the force and torque equations lead to more unknown forces than equations. Example: Four-legged table Detailed material properties and history determine the forces.

27 7/29/2004 Elasticity When a real object is subjected to a force (or a stress), it will deform (strain). Over a range of deformation, objects usually obey a form of Hooke’s law The “modulus” is like the spring constant. It varies for different materials and different stresses

28 7/29/2004 Types of Stress Tensile (and compressive): L F A L+  L E = Young’s modulus Not all materials are as good under tension as under compression (example: concrete)

29 7/29/2004 Types of Stress Shear stress: L A G = shear modulus xx F V VV Hydraulic stress: B = Bulk modulus p = pressure

30 7/29/2004 Elasticity Objects return to original shape if deformation is small enough. The point at which a deformation becomes permanent is called the yield strength The point at which on object breaks is called its ultimate strength

31 7/29/2004 Tensile Stress: