Static Equilibrium. Lecture 21. Chapter 12. Physics I Department of Physics and Applied Physics

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1 Lecture 21 Chapter 12 Physics I Static Equilibrium Course website: Lecture Capture:

2 Outline Chapter 12 The Conditions for Equilibrium Solving Statics Problems Stability and Balance

3 Exam III Info Exam III Wed Dec. 4, 9:00-9:50am, OH 150. Exam III covers Chapters 9-11 Same format as Exam II Prior Examples of Exam III posted Ch. 9: Linear Momentum (no section 10) Ch. 10: Rotational Motion (no section 10) Ch. 11: Angular Momentum; General Rotation (no sections 7-9) Exam Review Session TBA, Ball 210

4 Conservation of Angular Momentum. Kepler s second law Kepler s second law states that each planet moves so that a line from the Sun to the planet sweeps out equal areas in equal times. Use conservation of angular momentum to show this.

Let s find area swept by r in dt: da 1 2 r( vdt)sin( ) 1 2 rvdt sin da mrvsin da dt L dt 2m 2m Let s show that L

5 Let s find ang. moment. of the Earth (treat the Earth as a point): L r p L rpsin mrvsin Kepler s second law origin Next step. Let s find area swept by r in dt: da 1 2 r( vdt)sin( ) 1 2 rvdt sin da mrvsin da dt L dt 2m 2m Let s show that L is conserved: dl dt net r So if L is constant, da/dt is a constant Kepler s 2 nd law is a consequence of conservation of ang. momentum! ext since net 0, then r F L r F g v, vdt F const g p ( rˆ) m dl vdt sin( ) 0

ConcepTest 1 A figure skater spins with her arms extended. When she pulls in her arms, she reduces her rotational inertia and spins faster so that her angular momentum is conserved.

6 ConcepTest 1 A figure skater spins with her arms extended. When she pulls in her arms, she reduces her rotational inertia and spins faster so that her angular momentum is conserved. Compared to her initial rotational kinetic energy, her rotational kinetic energy after she pulls in her arms must be: K rot 2 I 2 ( I) 2 use L I K rot L 2 Because L is conserved, larger means larger K rot. The extra energy comes from the work she does on her arms. Figure Skater A) the same B) larger because she s rotating faster C) smaller because her rotational inertia is smaller

7 Bicycle wheel precession. A bicycle wheel is spun up to high speed and is suspended from the ceiling by a wire attached to one end of its axle. Now mg try to tilt the axle downward. You expect the wheel to go down (and it would if it weren t rotating), but it unexpectedly swerves to the left and starts rotating! Let s explain that. mg

wire F T x r y mg Bicycle wheel precession.

changes direction of angular momentum L I then angular

8 wire F T x r y mg Bicycle wheel precession. L fin dl L in Since z mg produces torque in +y direction r mg Apply Rotational N.2nd law: L fin L in dl dt dl rmg( ˆ) j dl dt L in dt Torgue changes direction of angular momentum L I then angular velocity changes its direction It is called precession Suitcase with a gyroscope

9 There are three branches of Mechanics: Kinematics Motion Forces Dynamics Motion Forces Statics Motion Forces

10 The 1st Condition for Equilibrium prevents translational motion N. 2 nd law describes translational motion F ma He doesn t want to have any sliding of a ladder, i.e. a 0 F 0 x y z F 0 F 0 F 0 An object with forces acting on it, but with zero net force, is said to be in equilibrium.

11 The 2nd Condition for Equilibrium prevents rotational motion Rotational N. 2 nd law describes rotational motion: x v ext I He doesn t want to have any rotation of a ladder, i.e. 0 ext 0 There must be no net external torque around any axis (the choice of axis is arbitrary). y 0 0 z 0

(B) The object is in torque equilibrium but not force equilibrium (C) The object is in both force equilibrium and torque equilibrium (D) The object is in neither

12 ConcepTest 2 Static equilibrium Consider a light rod subject to the two forces of equal magnitude as shown in figure. Choose the correct statement with regard to this situation: (A) The object is in force equilibrium but not torque equilibrium. (B) The object is in torque equilibrium but not force equilibrium (C) The object is in both force equilibrium and torque equilibrium (D) The object is in neither force equilibrium nor torque equilibrium F F F ext m 0 force equilibrium torque equilibrium X Here, the 1st condition is satisfied but the 2 nd isn t, so there will be rotation. So, to have static situation, both conditions must be satisfied.

13 Reduce # of Equilibrium Equations For simplicity, we will restrict the applications to situations in which all the forces lie in the xy plane. 1 st condition: x F 0 F 0 F 0 2 nd condition: x y z x y z There are three resulting equations, which we will use 1) F 0 2) F 0 3) 0 y z

14 Axis of rotation for the 3 rd equation z 0 Does it matter which axis you choose for calculating torques? NO. The choice of an axis is arbitrary If an object is in translational equilibrium and the net torque is zero about one axis, then the net torque must be zero about any other axis F 0 We should be smart to choose a rotation axis to simplify problems Any axis of rotation works

15 Any axis of rotation works for the 3 rd equation (proof) The choice of an axis is arbitrary z 0

16 Concurrent/Nonconcurrent forces F F 2 1 F 3 Concurrent forces: when the lines of action of the forces intersect at a common point, there will be no rotation. So z F F F 0 F x y x F 3 Nonconcurrent forces: when the lines of action of the forces do not intersect at a common point, there will be rotation. So z F F 0 y

17 Example:traffic light Find the tension in the two wires supporting the traffic light

18 Torque due to gravity Here, we will often have objects in which there is a torque exerted by gravity. There is a simple rule how to get gravitational torque: R CM Mg MgRsin R CM CM R CM As if the whole mass were here W Mg Mg For extended objects, gravitational torque acts as if all mass were concentrated at the center of mass

19 1. Choose one object at a time, and make a free-body diagram by showing all the forces on it and where they act. 2. Choose a coordinate system and resolve forces into components. 3. Write equilibrium equations for the forces. 4. Choose any axis perpendicular to the plane of the forces and write the torque equilibrium equation. A clever choice here can simplify the problem enormously. 5. Solve. Solving Statics Problems

20 Thank you See you on Monday

Conservation of Angular momentum

ecture 20 Chapter 11 Physics I 04.14.2014 Conservation of Angular momentum Course website: http://faculty.uml.edu/andriy_danylov/teaching/physicsi ecture Capture: http://echo360.uml.edu/danylov2013/physics1spring.html