1. Physics 201: Lecture 21, Pg 1 Lecture 21 Goals: Discussion conditions for static equilibrium Use Free Body Diagrams prior to problem solving Introduce Young’s, Shear and Bulk modulus Exam 3: Wednesday, April, 18 th 7:15-8:45 PM

2. Physics 201: Lecture 21, Pg 2 Statics Equilibrium is established when In 3D this implies SIX expressions (x, y & z)

3. Physics 201: Lecture 21, Pg 3 Three force and three torque expressions

4. Physics 201: Lecture 21, Pg 4 Free Body Diagram Rules l For forces zero acceleration reflects the motion of the center of mass l For torques the position of the force vector matters l We are free to choose the axis of rotation (usually one that simplifies the algebra)

5. Physics 201: Lecture 21, Pg 5 Statics: Using Torque l Now consider a plank of mass M suspended by two strings as shown. l We want to find the tension in each string: L/2 L/4 M x CM T1T1 T2T2 Mg y x

6. Physics 201: Lecture 21, Pg 6 Statics: Using Torque l Now consider a plank of mass M suspended by two strings as shown. l We want to find the tension in each string: L/2 L/4 M x CM T1T1 T2T2 Mg y x

7. Physics 201: Lecture 21, Pg 7 Statics: Using Torque l Now consider a plank of mass M suspended by two strings as shown. l Using a different torque position L/2 L/4 M x CM T1T1 T2T2 Mg y x

8. Physics 201: Lecture 21, Pg 8 Another Example: See-saw l Two children (60 kg and 30 kg) sit on a 3.0 m long uniform horizontal teeter-totter of mass 30 kg. The larger child is 1.0 m from the pivot point while the smaller child is trying to figure out where to sit so that the teeter-totter remains motionless. l Position of the small boy? 30 kg 60 kg 30 kg

9. Physics 201: Lecture 21, Pg 9 Example: Soln. l Draw a Free Body diagram (assume g = 10 m/s 2 ) l 0 = 300 d + 300 x 0.5 + N x 0 – 600 x 1.0 0= 2d + 1 – 4 d = 1.5 m from pivot point 30 kg 1 m 60 kg 30 kg 0.5 m 300 N 600 N N d

10. Physics 201: Lecture 21, Pg 10 Exercise: Statics l A 1.0 kg ball is hung at the end of a uniform rod exactly 1.0 m long. The system balances at a point on the rod 0.25m from the end holding the mass.  What is the mass of the rod? (a) 0.5 kg (b) 1 kg (c) 2 kg 1kg 1m

11. Physics 201: Lecture 21, Pg 11 The classic ladder problem A uniform ladder of length L and mass m is propped up against a frictionless wall. There is friction with the ground with static coefficient . The angle the ladder makes with the ground is . What is the minimum value of  to keep the latter from slipping? N N’N’ mg f  =0

12. Physics 201: Lecture 21, Pg 12 The classic ladder problem A uniform ladder of length L and mass m is propped up against a frictionless wall. There is friction with the ground with static coefficient . If the angle the ladder makes with the ground is . What is the minimum value of  that keeps the ladder from slipping? N N’N’ mgmg f  =0

13. Physics 201: Lecture 21, Pg 13 Example: Statics l A box is placed on a ramp in the configurations shown below. Friction prevents it from sliding. The center of mass of the box is indicated by a white dot in each case.  In which case(s) does the box tip over ? (a) all (b) 2 & 3 (c) 3 only 1 2 3

14. Physics 201: Lecture 21, Pg 14 A stable equilibrium l Potential energy with tipping (left and right) l x-axis corresponds to the x-position of the center-of mass 1

15. Physics 201: Lecture 21, Pg 15 Balancing Acts l So, where is the center of mass for this construction? l Between the arrows!

16. Physics 201: Lecture 21, Pg 16 Example problem The board in the picture has an average length of 1.4 m, mass 2.0 kg and width 0.14 m. The angle of the board is 45 °. The bottle is 1.5 m long, a center of mass 2/5 of the way from the bottom and is mounted 1/5 of way from the end. If the sketch represents the physics of the problem. l What are the minimum and maximum masses of the bottle that will allow this system to remain in balance? -0.1 m -0.5 m -0.8 m 0.1 m 0.0 m 0.4 m

17. Physics 201: Lecture 21, Pg 17 Example problem l What are the minimum and maximum masses of the bottle that will allow this system to remain in balance? l X center of mass, x, -0.1 m < x < 0.1 m -0.1 m -0.5 m -0.8 m 0.1 m 0.0 m 0.4 m 2.0 kg

18. Physics 201: Lecture 21, Pg 18 Example problem l What are the minimum and maximum masses of the bottle that will allow this system to remain in balance? l Check torque condition (g =10 m/s 2 ) l x, -0.1 m < x < 0.1 m -0.1 m -0.5 m -0.8 m 0.1 m 0.0 m 0.4 m 2.0 kg

19. Physics 201: Lecture 21, Pg 19 Real Physical Objects l Representations of matter  Particles: No size, no shape  can only translate along a path  Extended objects are collections of point-like particles They have size and shape so, to better reflect their motion, we introduce the center of mass Rigid objects: Translation + Rotation Deformable objects: Regular solids: Shape/size changes under stress (applied forces) Reversible and irreversible deformation Liquids: They do not have fixed shape Size (aka volume) will change under stress

20. Physics 201: Lecture 21, Pg 20 For Thursday l All of chapter 12

21. Physics 201: Lecture 21, Pg 21 l Young’s modulus: measures the resistance of a solid to a change in its length. l Bulk modulus: measures the resistance of solids or liquids to changes in their volume. Some definitions L0L0 LL F V0V0 V 0 -  V F l Elastic properties of solids : elasticity in length volume elasticity